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1936 in sports

1936 in sports describes the year's events in world sport.

Alpine skiing
FIS Alpine World Ski Championships 6th FIS Alpine World Ski Championships are held at Innsbruck, Austria. The events are a downhill, a slalom and a combined race in both the men's and women's categories. The winners are:
 * Men's Downhill – Rudolf Rominger (Switzerland)
 * Men's Slalom – Rudolph Matt (Austria)
 * Men's Combined – Rudolf Rominger (Switzerland)
 * Women's Downhill – Evelyn Pinching (Great Britain)
 * Women's Slalom – Gerda Paumgarten (Austria)
 * Women's Combined – Evelyn Pinching (Great Britain)

American football

 * NFL Championship: the Green Bay Packers won 21–6 over the Boston Redskins. The game was moved from Boston to New York's Polo Grounds due to fan apathy.
 * Rose Bowl (1935 season):
 * The SMU Mustangs lose 7–0 to the Stanford Indians; share national championship
 * Minnesota Golden Gophers – college football national championship; first season in which a national champion was declared by the AP Poll.
 * First NFL draft held
 * Second American Football League (AFL) founded
 * Cleveland Rams founded by Homer Marsh as part of AFL

Association football
England Spain Germany Italy Portugal France Ukraine
 * First Division – Sunderland win the 1935–36 title.
 * FA Cup – Arsenal beat Sheffield United 1–0, thanks to a Ted Drake goal.
 * La Liga won by Athletic Bilbao
 * German football championship won by Nürnberg
 * Serie A won by Bologna
 * Primeira Liga won by S.L. Benfica
 * French Division 1 won by RC Paris
 * FC Shakhtar Donetsk was founded in May (Former part of Soviet Union)

Australian rules football

 * Victorian Football League
 * Collingwood wins the 40th VFL Premiership, beating South Melbourne 11.23 (89) to 10.18 (78) in the 1936 VFL Grand Final.
 * Brownlow Medal awarded to Dinny Ryan (Fitzroy)
 * South Australian National Football League
 * 1 August: Glenelg 14.9 (93) defeat South Adelaide 9.32 (86) despite having 23 scoring shots against South's 41. The eighteen-shot deficit is the greatest deficit in scoring shots by a winning team in a major Australian Rules league.
 * 3 October: Port Adelaide 13.19 (97) defeat Sturt 14.10 (94) for their first premiership since 1929 and their eleventh overall. It is the only SANFL Grand Final where the winning team scored fewer goals than the losing team.
 * Magarey Medal won by Bill McCallum (Norwood)
 * Western Australian National Football League
 * 10 October: East Perth 11.5 (71) defeat Claremont 9.6 (60) to win their eighth WANFL premiership.
 * Sandover Medal won by George Moloney (Claremont)

Baseball

 * Japanese Baseball League, as predecessor for Japan Central League and Pacific League of Japan, a first officially game held on February 9.
 * Plans are announced for a Baseball Hall of Fame to be established in 1939, the game's supposed centennial, in Cooperstown, New York. In the first elections to select 15 initial inductees (5 from the 19th century and 10 from the 20th), Ty Cobb, Babe Ruth, Honus Wagner, Christy Mathewson and Walter Johnson are selected from the 20th century; the election for 19th century players is plagued by problems and results in no selections. See: Baseball Hall of Fame balloting, 1936
 * World Series – New York Yankees defeat the New York Giants, 4–2.
 * Nagoya Baseball Club, as predecessor for Chunichi Dragons, officially founded in Japan on January 15.

Basketball

 * August 14 – 1936 Summer Olympics: The United States men's national basketball team wins the first Olympic basketball tournament in the final game over Canada, 19–8.
 * The first season for Argentine Basketball Club Championship League was held, Huracan de Rosario has first season's champion.

Boxing
Events Lineal world champions
 * 19 June – in one of boxing's biggest-ever upsets, Max Schmeling knocks out Joe Louis at 2:29 of round 12 at New York's Yankee Stadium
 * World Heavyweight Championship – James J. Braddock
 * World Light Heavyweight Championship – John Henry Lewis
 * World Middleweight Championship – vacant
 * World Welterweight Championship – Barney Ross
 * World Lightweight Championship – Tony Canzoneri → Lou Ambers
 * World Featherweight Championship – vacant
 * World Bantamweight Championship – Sixto Escobar → Tony Marino → Sixto Escobar
 * World Flyweight Championship – vacant → Benny Lynch

Cricket
Events England Australia India New Zealand South Africa West Indies
 * Australia tours South Africa, winning the five-test series three tests to nil.
 * County Championship – Derbyshire
 * Minor Counties Championship – Hertfordshire
 * Most runs – Patsy Hendren 2,654 @ 47.39 (HS 202)
 * Most wickets – Hedley Verity 216 @ 13.18 (BB 9–12)
 * India play a three-Test series of England, losing two and drawing one
 * Wisden Cricketers of the Year – Charlie Barnett, Bill Copson, Alf Gover, Vijay Merchant, Stan Worthington
 * Sheffield Shield – South Australia
 * Most runs – Don Bradman 1,173 @ 130.33 (HS 369)
 * Most wickets – Frank Ward 50 @ 20.94 (BB 6–47)
 * Ranji Trophy – Bombay beat Madras by 190 runs
 * Bombay Quadrangular – Hindus
 * Plunket Shield – Wellington
 * Currie Cup – Western Province
 * Inter-Colonial Tournament – British Guiana

Cycling
Tour de France Giro d'Italia
 * Sylvère Maes wins the 30th Tour de France
 * Gino Bartali of Legnano wins the 24th Giro d'Italia

Field Hockey

 * Olympic Games (Men's Competition) in Berlin won by India

Figure skating

 * World Figure Skating Championships:
 * Men's singles: Karl Schäfer
 * Ladies' singles: Sonja Henie
 * Pairs: Maxi Herber and Ernst Baier
 * 1936 Winter Olympics
 * Men's singles: Karl Schäfer
 * Ladies' singles: Sonja Henie
 * Pairs: Maxi Herber and Ernst Baier

Golf
Men's professional Men's amateur Women's professional
 * Masters Tournament – Horton Smith
 * U.S. Open – Tony Manero
 * British Open – Alf Padgham
 * PGA Championship – Denny Shute
 * British Amateur – Hector Thomson
 * U.S. Amateur – Johnny Fischer
 * Women's Western Open – Opal Hill

Horse racing
Steeplechases Flat races
 * May 8 – jockey Ralph Neves was involved in a racing accident at Bay Meadows Racetrack in San Mateo, California and mistakenly pronounced dead. A while later, he woke up in the morgue and promptly returned to the racetrack but was not allowed to compete in any of the remaining races because of his "death".
 * Cheltenham Gold Cup – Golden Miller
 * Grand National – Reynoldstown
 * Australia – Melbourne Cup won by Wotan
 * Canada – King's Plate won by Monsweep
 * France – Prix de l'Arc de Triomphe won by Corrida
 * Ireland – Irish Derby Stakes won by Raeburn
 * English Triple Crown Races:
 * 2,000 Guineas Stakes – Pay Up
 * The Derby – Mahmoud
 * St. Leger Stakes – Boswell
 * United States Triple Crown Races:
 * Kentucky Derby – Bold Venture
 * Preakness Stakes – Bold Venture
 * Belmont Stakes – Granville

Ice hockey

 * The Detroit Red Wings win the Stanley Cup Championship 3 games to 1 over the Toronto Maple Leafs
 * Combining this with the Detroit Tigers World Series win and the Detroit Lions NFL Championship, in 1935, Detroit played home to the Championship teams in the MLB, NFL, and NHL in one 12-month period. This feat has yet to be duplicated by any other city.
 * The first season of Czechoslovak Extraliga was held, which separated into Czech Extraliga and Tipsport Ice-hockey Liga of Slovakia from this league on 1993.

Olympic Games

 * The infamous 1936 Summer Olympics take place in Berlin
 * Germany wins the most medals (89) and the most gold medals (33)
 * Jesse Owens wins four athletics gold medals
 * The first torch relay takes place from Olympia, Greece
 * 1936 Winter Olympics takes place in Garmisch-Partenkirchen, Germany
 * Norway wins the most medals (15) and the most gold medals (7)

Rowing
The Boat Race
 * 4 April — Cambridge wins the 88th Oxford and Cambridge Boat Race

Rugby league

 * 1935–36 European Rugby League Championship / 1936–37 European Rugby League Championship
 * 1936 New Zealand rugby league season
 * 1936 NSWRFL season
 * 1936–37 Northern Rugby Football League season / 1935–36 Northern Rugby Football League season

Rugby union

 * 49th Home Nations Championship series is won by Wales

Snooker

 * World Snooker Championship – Joe Davis beats Horace Lindrum 34–27

Speed skating
Events Speed Skating World Championships 1936 Winter Olympics (Men) 1936 Winter Olympics (Women)
 * Inaugural World Allround Speed Skating Championships for Women is held in Stockholm
 * Men's All-round Champion – Ivar Ballangrud (Norway)
 * Women's All-round Champion – Kit Klein (USA)
 * 500m – gold medal: Ivar Ballangrud (Norway)
 * 1500m – gold medal: Charles Mathiesen (Norway)
 * 5000m – gold medal: Ivar Ballangrud (Norway)
 * 10000m – gold medal: Ivar Ballangrud (Norway)
 * not contested

Tennis
Australia England France USA Davis Cup
 * Australian Men's Singles Championship – Adrian Quist (Australia) defeats Jack Crawford (Australia) 6–2, 6–3, 4–6, 3–6, 9–7
 * Australian Women's Singles Championship – Joan Hartigan Bathurst (Australia) defeats Nancye Wynne Bolton (Australia) 6–4, 6–4
 * Wimbledon Men's Singles Championship – Fred Perry (Great Britain) defeats Gottfried von Cramm (Germany) 6–1, 6–1, 6–0
 * Wimbledon Women's Singles Championship – Helen Jacobs (USA) defeats Hilde Krahwinkel Sperling (Denmark) 6–2, 4–6, 7–5
 * French Men's Singles Championship – Gottfried von Cramm (Germany) defeats Fred Perry (Great Britain) 6–0, 2–6, 6–2, 2–6, 6–0
 * French Women's Singles Championship – Hilde Krahwinkel Sperling (Germany) defeats Simonne Mathieu (France) 6–3, 6–4
 * American Men's Singles Championship – Fred Perry (Great Britain) defeats Don Budge (USA) 2–6, 6–2, 8–6, 1–6, 10–8
 * American Women's Singles Championship – Alice Marble (USA) defeats Helen Jacobs (USA) 4–6, 6–3, 6–2
 * 1936 International Lawn Tennis Challenge – at 3–2  at Centre Court, Wimbledon (grass) London, United Kingdom

Awards

 * Associated Press Male Athlete of the Year – Jesse Owens, Track and field
 * Associated Press Female Athlete of the Year – Helen Stephens, Track and field

title: 1936 in sports


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Olympic National Forest

Olympic National Forest is a U.S. National Forest located in Washington, USA. With an area of 628115 acre, it nearly surrounds Olympic National Park and the Olympic Mountain range. Olympic National Forest contains parts of Clallam, Grays Harbor, Jefferson, and Mason counties. The landscape of the national forest varies, from the temperate Olympic rain forest to the salt water fjord of Hood Canal to the peaks of Mt. Washington.

Annual precipitation averages about 220 inches, giving rise to streams such as the Humptulips River.

Olympic National Forest was originally created as Olympic Forest Reserve in 1897, then renamed to Olympic National Forest in 1907. A portion of the National Forest became the Mount Olympus National Monument in 1909, which was later designated Olympic National Park.

A 1993 Forest Service study estimated that the extent of old growth in the Forest was 266800 acre. It is administered in two ranger districts: the Pacific Ranger District on the west side of the Olympic Peninsula, and the Hood Canal Ranger District on the east side.

Forest headquarters are located in Olympia, with ranger district offices in Forks, Quinault, and Quilcene. The former office in Hoodsport closed in 2005, and now houses a local Chamber of Commerce, which still sells Northwest Forest Passes.

Other Washington towns near entrances of the forest include Port Angeles, Sequim, and Amanda Park.

Points of interest

 * Lake Cushman
 * Quinault Rain Forest
 * Wynoochee Dam
 * Lake Crescent

Wilderness areas

 * The Brothers Wilderness
 * Buckhorn Wilderness
 * Colonel Bob Wilderness
 * Mt. Skokomish Wilderness
 * Wonder Mountain Wilderness

Wild Olympics
The Wild Olympics campaign is an effort to designate additional areas on the Olympic Peninsula as protected.

Under a bill introduced by United States Senator Patty Murray in January 2014 logging on an additional 126,554 acres (20%) of the Olympic National Forest's lands would be disallowed under the creation of nine new wilderness areas and expansion of the five existing ones. Wild and Scenic River designations would extend to 19 rivers, including those originating in the Olympic National Park, such as the Quinault, Hoh, Elwha, and Hamma Hamma, and some that do not, like the Middle Fork of the Satsop River. According to The Oregonian, "the wilderness designation would permanently protect old growth and ancient forest habitat throughout the region. The wild and scenic rivers designation would add federal recognition to the outstanding river systems on the peninsula, protecting them as a source of clean drinking water and helping to keep Puget Sound clean for generations. This designation does not restrict private property rights."

Like the Northern Spotted Owl controversy two decades earlier, the action has met opposition from some residents on and near the Olympic Peninsula who see it as a threat to their livelihoods. Handmade "Stop Wild Olympics" signs can be seen in the Aberdeen area and on the peninsula, expressing displeasure, and an opposition group, Working Wild Olympics, was set up in 2011. The effort has the support of dozens of "major hunting and fishing organizations" who see it as a way to sustain elk, salmon and other fish. Debate on the topic in 2014 has included community meetings drawing hundreds of individuals on both sides, and full-page advertisements in local newspapers.

An episode of This American Land concerning the Wild Olympics proposal showed nationally on PBS in 2014.

The Wild Olympics Wilderness and Wild and Scenic Rivers Act was reintroduced in 2019 as H.R.2642 and received a hearing in the House Natural Resources Subcommittee on National Parks, Forests, and Public Lands.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Olympic National Forest


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Mint

Mint or The Mint may refer to:

Plants

 * Lamiaceae, the mint family
 * Mentha, the genus of plants commonly known as "mint"

Coins and collectibles

 * Mint (facility), a facility for manufacturing coins
 * Mint condition, a state of like-new quality

Fictional entities

 * Mint, in the video game Threads of Fate
 * Mint, a Ranma ½ character
 * Mint Adnade, in the video game Tales of Phantasia
 * Mint Aizawa, in the anime and manga Tokyo Mew Mew
 * Mint Blancmanche, in the video game/anime series Galaxy Angel

Film and television

 * Mint (film), a Japanese drama
 * The Mint (film), a 2015 American comedy
 * The Mint (Australia game show), 2007–2008
 * The Mint (British game show), 2006–2007

Music

 * Mint (band), a Belgian music group
 * Mint Records, a record label
 * Mint (Alice Merton album), 2019
 * Mint, a 1983 album by Meiko Nakahara
 * Mint, a 2003 album by Alexkid in collaboration with Liset Alea
 * "Mint" (Namie Amuro song), 2016
 * "Mint" (Rina Aiuchi song), 2007
 * "Mint" (Lindsay Ell song), 2017
 * "Mint" (Misako Uno song), 2019
 * "The Mint", a song by Earl Sweatshirt from the 2018 album Some Rap Songs

Literature

 * Mint (newspaper), a business newspaper published in India
 * The Mint (book), by T. E. Lawrence, 1955

Businesses, organisations and products

 * Mint (restaurant), in Dublin, Ireland
 * Intuit Mint, formerly Mint.com, a personal financial management website and mobile app
 * Malaysian Institute of Nuclear Technology Research, now Malaysian Nuclear Agency
 * Minor International, a hospitality company, Stock Exchange of Thailand symbol MINT
 * Mint Airways, a former Spanish airline
 * MiNT camera, a company specializing in instant cameras and accessories
 * Mint Mobile, a mobile virtual network operator in the United States
 * Mint Productions, an Irish TV production company
 * The Mint Las Vegas, a hotel and casino in Las Vegas 1957–1989
 * JetBlue Mint, a premium cabin service of JetBlue
 * Mint, a robot by Evolution Robotics
 * Ministry of Innovation and Technology (Ethiopia)

Computing

 * Mint (web analytics software), discontinued 2016
 * MiNT, an operating system for the Atari ST
 * Linux Mint, a distribution of the Linux operating system

Places

 * Mint, Arizona, U.S.
 * Liberty of the Mint, or The Mint, a district in Southwark, London, England
 * River Mint, in Cumbria, England
 * Mint River, in Alaska, U.S.
 * The Mint (Carlingford), a fortified house and museum in Ireland
 * Sydney Mint, in New South Wales, Australia
 * Vallalar Nagar, Chennai, India, popularly called Mint

Other uses

 * Mint (candy), a food item flavoured with mint
 * MINT (economics), the economies of Mexico, Indonesia, Nigeria, and Turkey
 * Mint (singer) (born 1994), a Thai singer based in South Korea
 * Mint, a shade of the colour spring green

title: Mint


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User:Allenthar

The distribution $$F(x) =\prod_{i=1}^n F_{i}(x)$$ can be generated from $$F(x) = P(max(F_{1},...,F_{n}) \le k)$$ because the product of independent RV's are the same as the max of the RV's. If the max probability of all of the $$F_{i}$$'s is less than k, then all of the individual $$F_{i}$$'s must be less than k.

title: User:Allenthar


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Gifford Pinchot National Forest

Gifford Pinchot National Forest is a National Forest located in southern Washington, managed by the United States Forest Service. With an area of 1.32 million acres (5300 km2), it extends 116 km along the western slopes of Cascade Range from Mount Rainier National Park to the Columbia River. The forest straddles the crest of the South Cascades of Washington State, spread out over broad, old growth forests, high mountain meadows, several glaciers, and numerous volcanic peaks. The forest's highest point is at 12276 ft at the top of Mount Adams, the second tallest volcano in the state after Rainier. Often found abbreviated GPNF on maps and in texts, it includes the 110000 acre Mount St. Helens National Volcanic Monument, established by Congress in 1982.

History
Gifford Pinchot National Forest is one of the older national forests in the United States. Included as part of the Mount Rainier Forest Reserve in 1897, 941440 acre was set aside as the Columbia National Forest on July 1, 1908.

In 1855, the US government commissioned Washington Territory to negotiate land cession treaties with tribes around the forest. The Yakama tribe signed a treaty agreement that stipulated their moving to a reservation while maintaining off-reservation resource rights; however, the original treaty was then broken in 1916 when the Washington State Supreme Court ruled that Yakamas' hunting off the reservation had to subscribe to state fish and game laws. Many tribes in the area have continued to use the area's resources while encountering non-Native hunters, fishers, and recreation users.

It was later renamed the Gifford Pinchot National Forest on June 15, 1949, in honor of Gifford Pinchot, one of the leading figures in the creation of the national forest system of the United States. His widow and fellow conservationist, Cornelia Bryce Pinchot, was one of the speakers who addressed the audience assembled that day. In 1985 the non-profit Gifford Pinchot Task Force formed to promote conservation of the forest.

Geography
Gifford Pinchot National Forest is located in a mountainous region approximately between Mount St. Helens to the west, Mount Adams to the east, Mount Rainier National Park to the north, and the Columbia River to the south. This region of Southwest Washington is noted for its complex topography and volcanic geology. About 65 percent of the forest acreage is located in Skamania County. In descending order of land area the others are Lewis, Yakima, Cowlitz, and Klickitat counties.

Major rivers
The Pacific Northwest brings abundant rainfall to the Gifford Pinchot National Forest, feeding an extensive network of rivers. The forest has only one river currently designated as Wild and Scenic, the White Salmon River, fed from glaciers high on Mount Adams. The Gifford Pinchot National Forest recommends four rivers to be added to the Wild and Scenic System. They are the Lewis River, the Cispus River, the Clear Fork and the Muddy Fork of the Cowlitz River. There are an additional thirteen rivers in the forest being studied for consideration into the national Wild and Scenic River System.

The following listed are the major streams and rivers of the Gifford Pinchot National Forest. Many of these provide excellent fishing.
 * Cispus River
 * Cowlitz River
 * White Salmon River
 * Little White Salmon River
 * Wind River
 * Lewis River
 * Muddy River
 * East Canyon Creek
 * Skate Creek
 * Iron Creek
 * Trout Lake Creek
 * Cultus Creek
 * Quartz Creek
 * Butter Creek
 * Clear Creek
 * Siouxon Creek
 * Canyon Creek
 * Johnson Creek

Major lakes
The Gifford Pinchot National Forest includes many popular and secluded backcountry lakes. Most of the lakes offers excellent fishing. Goose Lake is known for the best fishing in the State of Washington.

The following table lists the major lakes of the Gifford Pinchot National Forest:

Congressional action
Congressional action since 1964 has established one national monument and seven wilderness areas in the Gifford Pinchot National Forest.

National Monuments
On August 26, 1982, congressional action established the Mount St. Helens National Volcanic Monument, after the cataclysmic eruption of Mount St. Helens in 1980.

Wilderness Areas
Congressional action since 1964 has established the following wilderness areas:
 * Glacier View Wilderness – 3060 acre
 * Goat Rocks – 108020 acre, part of which lies in neighboring Wenatchee NF
 * Indian Heaven – 20,782 acres (84.1 km2)
 * Mount Adams – 47,078 acres (190.5 km2)
 * Tatoosh – 15,725 acres (63.6 km2)
 * Trapper Creek – 5,969 acres (24.2 km2)
 * William O. Douglas – 168955 acre, most of which lies in neighboring Wenatchee NF
 * The Shark Rock Wilderness was proposed in the mid-1970s by E.M. Sterling for the 75,000 acre Dark Divide Roadless Area in his book, The South Cascades. It is the largest unprotected roadless area (allowing motorized access) in the Washington Cascades, featuring sharp peaks and deep canyons, old growth forests, and delicate subalpine meadows.

Points of interest
The forest also offers the following special areas and points of interest:
 * Dark Divide Roadless Area
 * Silver Star Scenic Area
 * Lava tubes, caves, and casts (notably the Ice Caves)
 * Ape Caves
 * Midway High Lakes Area
 * Big Lava Bed
 * Packwood Lake
 * Sawtooth Berry Fields, huckleberry fields reserved for Yakima tribe use. Designated in 1932 through a handshake agreement between forest supervisor J.R. Bruckart and Yakima Chief William Yallup.
 * Lone Butte Wildlife Emphasis Area
 * Layser Cave, a dwelling for Native Americans 7,000 years ago and a former archeological site.

Forest Service management
The forest supervisor's office is located in Vancouver, Washington. There are local ranger district offices in Randle, Amboy, and Trout Lake. The forest is named after the first chief of the United States Forest Service, Gifford Pinchot. Washington towns near entrances of the forest include Cougar, Randle, Packwood, Trout Lake, and Carson.

Ecology
A 1993 Forest Service study estimated that the extent of old growth in the Forest was 198000 acre, some of which is contained within its wilderness areas.

The Gifford Pinchot National Forest is the native habitat for several threatened species which include the spotted owl (threatened 2012) as well as multiple species of Northwest fish like the bull trout (threatened 1998), chinook salmon (threatened 2011), coho salmon (threatened 2011) and steelhead (threatened 2011).

People for over 6,000 years have made an impact in the ecology of the Gifford Pinchot National Forest. Native Americans hunted in high meadows below receding glaciers. The natives then began to manage the forest to meet their own needs. One method they used was to burn specific areas to help in the huckleberry production. About 338 spots more than 6,000 culturally modified trees were identified, of which 3,000 are protected now. Archaeological investigations are supported by the United States Forest Service.

The forest was home to the Big Tree at the southern flank of Mt Adams, one of the world's largest Ponderosa Trees.

title: Gifford Pinchot National Forest


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Poncelet–Steiner theorem

In the branch of mathematics known as Euclidean geometry, the Poncelet–Steiner theorem is one of several results concerning compass and straightedge constructions having additional restrictions imposed on the traditional rules. This result, related to the rusty compass equivalence, states that whatever can be constructed by straightedge and compass together can be constructed by straightedge alone, provided that a single circle and its centre are given:

 * Any Euclidean construction, insofar as the given and required elements are points (or lines), if it can be completed with both the compass and the straightedge together, may be completed with the straightedge alone provided that no fewer than one circle with its center exist in the plane.

Though a compass can make constructions significantly easier, it is implied that there is no functional purpose of the compass once the first circle has been drawn. All constructions remain possible, though it is naturally understood that circles and their arcs cannot be drawn without the compass. All points that uniquely define a construction, which can be determined with the use of the compass, are equally determinable without, albeit with greater difficulty.

This means only that the compass may be used for aesthetic purposes, rather than for the purposes of construction. In other words, the compass may be used after all of the key points are determined, in order to "fill-in" the arcs purely for visual or artistic purposes, if it is desirable, and not as a necessary step toward construction. Nothing is lost by neglecting the construction of circular arcs.

Constructions carried out in adherence with this theorem - relying solely on the use of a straightedge tool without the aid of a compass - are known as Steiner constructions. Steiner constructions may involve any number of circles, including none, already drawn in the plane, with or without their centers. They may involve all manner of unique shapes and curves preexisting in the plane, also, provided that the straightedge tool is the only physical tool at the geometers disposal. Whereas the Poncelet-Steiner theorem stipulates the existence of a circle and its center, and affirms that a single circle is equivalent to a compass.

History

In the tenth century, the Persian mathematician Abu al-Wafa' Buzjani (940−998) considered geometric constructions using a straightedge and a compass with a fixed opening, a so-called rusty compass. Constructions of this type appeared to have some practical significance as they were used by artists Leonardo da Vinci and Albrecht Dürer in Europe in the late fifteenth century. A new viewpoint developed in the mid sixteenth century when the size of the opening was considered fixed but arbitrary and the question of how many of Euclid's constructions could be obtained was paramount.

Renaissance mathematician Lodovico Ferrari, a student of Gerolamo Cardano in a "mathematical challenge" against Niccolò Fontana Tartaglia was able to show that "all of Euclid" (that is, the straightedge and compass constructions in the first six books of Euclid's Elements) could be accomplished with a straightedge and rusty compass. Within ten years additional sets of solutions were obtained by Cardano, Tartaglia and Tartaglia's student Benedetti. During the next century these solutions were generally forgotten until, in 1673, Georg Mohr published (anonymously and in Dutch) Euclidis Curiosi containing his own solutions. Mohr had only heard about the existence of the earlier results and this led him to work on the problem.

Showing that "all of Euclid" could be performed with straightedge and rusty compass is not the same as proving that all straightedge and compass constructions could be done with a straightedge and just a rusty compass. Such a proof would require the formalization of what a straightedge and compass could construct. This groundwork was provided by Jean Victor Poncelet in 1822, having been motivated by Mohr's work on the Mohr-Mascheroni theorem. He also conjectured and suggested a possible proof that a straightedge and rusty compass would be equivalent to a straightedge and compass, and moreover, the rusty compass need only be used once. The result of this theorem, that a straightedge and single circle with given centre is equivalent to a straightedge and compass was proved by Jakob Steiner in 1833.

Relationships to other constructs
Various other notions, tools, terminology, etc., is often associated (sometimes loosely) to the Poncelet-Steiner theorem. Some are listed here.

Steiner constructions
The term Steiner construction typically refers to any geometric construction that utilizes the straightedge tool only, and is sometimes simply called a straightedge-only construction. No stipulations are made about what geometric objects already exist in the plane or their relative placement; any such conditions are postulated ahead of time. Also, no implications are made about what is or is not possible to construct.

Therefore, all constructions adhering to the Poncelet-Steiner theorem are Steiner constructions, though not all Steiner constructions abide by the condition of only one circle with its center provided in the plane. The Poncelet-Steiner theorem does not require an actual compass - it is presumed that the circle preexists in the plane - therefore all constructions herein demonstrating the Poncelet-Steiner theorem are Steiner constructions.

Rusty compass
The rusty compass describes a compass whose hinge is so rusted as to be fused such that its legs - the needle and pencil - are unable to adjust width. In essence, it is a compass whose distance is fixed, and which draws circles of a predetermined and constant, but arbitrary radius. Circles may be drawn centered at any arbitrary point, but the radius is unchangeable.

As a restricted construction paradigm, the rusty compass constructions allow the use of a straightedge and the fixed-width compass. The rusty compass equivalence:

 * All points necessary to uniquely describe any compass-straightedge construction may be achieved with a straightedge and fixed-width compass.

It is naturally understood that the arbitrary-radius compass may be used for aesthetic purposes; only the arc of the fixed-width compass may be used for construction.

In some sense, the rusty compass is a generalization and simplification of the Poncelet-Steiner theorem. Though not more powerful, it is certainly more convenient. The Poncelet-Steiner theorem requires a single circle with arbitrary radius and center point to be placed in the plane. As it is the only drawn circle, whether or not it was drawn by a rusty compass is immaterial and equivalent. The benefit of general rusty compass constructions, however, is that the compass may be used repeatedly to redraw circles centered at any desired point, albeit with the same radius, thus simplifying many constructions. Naturally if all constructions are possible with a single circle arbitrarily placed in the plane, then the same can surely be said about a straightedge and rusty compass, with which at least one circle may be arbitrarily placed.

It is known that a straightedge and a rusty compass is sufficient to construct all that is possible with straightedge and standard compass - with the implied understanding that circular arcs of arbitrary radii cannot be drawn, and only need be drawn for aesthetic purposes rather than constructive ones. Historically this was proven when the Poncelet-Steiner theorem was proven, which is a stronger result. The rusty compass, therefore, is no weaker than the Poncelet-Steiner theorem. The rusty compass is also no stronger.

The Poncelet-Steiner theorem reduces Ferrari's rusty compass equivalence, a claim at the time, to a single-use compass:
 * All points necessary to uniquely describe any compass-straightedge construction may be achieved with only a straightedge, once the first circle has been placed.

The Poncelet-Steiner theorem takes the rusty compass scenario, and breaks the compass completely after its first use.

Steiner's theorem / Hilbert's error

 * Not to be confused for Steiner's parallel axis theorem, Steiner's porism, or the Steiner–Lehmus_theorem''.

If only one circle is to be given and no other special information, Steiner's theorem implies that the center of the circle must be provided along with the arc of the circle. This is done by proving the impossibility of constructing the circle's center from straightedge alone using only a single circle in the plane, without its center. An argument using projective transformations and Steiner's conic sections is used.

 *  With only one circle provided in the plane, its center cannot be constructed by straightedge alone.

Also attributed to David Hilbert and known as Hilbert's Error, a naïve summary of the proof is as follows. With the use of a straightedge tool, only linear projective transformations are possible, and linear projective transformations are reversible operations. Lines project onto lines under any linear projective transformation, while conic sections project onto conic sections under a linear projective transformation, but the latter are skewed such that eccentricities, foci, and centers of circles are not preserved. Under different sequences of mappings, the center does not map uniquely and reversibly. This would not be the case if lines could be used to determine a circles center. As linear transformations are reversible operations and would thus produce unique results, the fact that unique results are not possible implies the impossibility of center-point constructions. The uniqueness of the constructed center would depend on additional information - not provided by a single circle - which would make the construction reversible.

Thus it is not possible to construct everything that can be constructed with straightedge and compass with straightedge alone. Consequently, requirements on the Poncelet-Steiner theorem cannot be weakened with respect to the circle center. If the centre of the only given circle is not provided, it cannot be obtained by a straightedge alone. Many constructions are impossible with straightedge alone. Something more is necessary, and a circle with its center identified is sufficient.

Alternative Frameworks to the Single Circle with Center
Alternatively, the center may be omitted with sufficient additional information. This is not a weakening of the Poncelet-Steiner theorem, merely an alternative framework. Nor is it a contradiction of Steiner's Theorem which hypothesizes only a single circle. The inclusion of this sufficient alternative information, which always includes at least two circles, disambiguates the mappings under the projective transformations, thus allowing various Steiner constructions to recover the circle center.

Each of these alternatives requires at least two circles devoid of their centers, plus some other unique piece of information. Some alternatives include two concentric or two intersecting circles, or three circles, or other variations wherein the provided circles are devoid of their centers. In each, some additional unique-but-sufficient criterion is met, such as concentricity, intersection points, a third circle, etc., respectively. In any of these cases, the center of a circle can be constructed, thereby reducing the problem to the Poncelet-Steiner theorem hypothesis (with the added convenience of having additional circles in the plane, all of whose centers may now be constructed).

Constructive proof outline
To prove the theorem, each of the basic constructions of compass and straightedge need to be proven to be possible by using a straightedge alone (provided that a circle and its center exist in the plane), as these are the foundations of, or elementary steps for, all other constructions. That is to say, all constructions can be written as a series of steps involving these five basic constructions:
 * 1) Creating the line through two existing points
 * 2) Creating the circle through one point with centre another point
 * 3) Creating the point which is the intersection of two existing, non-parallel lines
 * 4) Creating the one or two points in the intersection of a line and a circle (if they intersect)
 * 5) Creating the one or two points in the intersection of two circles (if they intersect).

#1 – A line through two points

This can be done with a straightedge alone. Neither a compass nor a circle is required.

#2 – A circle through one point with defined center

It is understood that the arc of a circle cannot be drawn without a compass. A circle is considered to be given by any two points, one defining the center and one existing on the circumference at radius. Any such pair define a unique circle, although the converse is not true: for any given circle there is no unique pair defining it. In keeping with the intent of the theorem which we aim to prove, the actual circle need not be drawn but for aesthetic reasons.

#3 – Intersection of two lines

This construction can also be done directly with a straightedge. #4, #5 – Intersections involving the circle

Thus, to prove the theorem, only constructions #4 and #5 need be proven possible using only a straightedge and a given circle with its center.

Notes and Caveats for the Constructive Proof
Usability

The constructive proof does not merely serve as a proof of the theorem, but also demonstrates the practical application of the most basic constructions, such that the claim of constructability with a straightedge could be employed in practice, in the most general case. Since all geometric constructions can be expressed as a sequence of the five basic constructive steps, and the below constructions demonstrate and justify each of these, necessarily, in order to prove the theorem, therefore all possible constructions may be implemented accordingly.

Doubts about Constructions #1, #2, or #3

Any doubts about constructions #1 or #3 would apply equally to the traditional construction paradigm involving the compass, and thus are not concerns unique to the Poncelet-Steiner theorem. Their justifications - if any need be given - will not be explored in this article.

Construction #2 should not be of concern. Even though it is undisputed that a unique circle is defined by a center point and a point on its circumference, the pertinent question is whether or not this is sufficient information for the purposes of straightedge-only construction, or if the drawn arc is required. The arc of the circle is only used in traditional construction paradigms for the purposes of circle-circle and circle-line intersections, therein the arc of the circle is used directly to identify intersection points. Thus if constructions #4 and #5 are satisfiable without the arc of the circle with which to intersect, then it will prove the non-necessity of drawing the arc. This would therefore imply that construction #2 is satisfied by a simple labeling of two points, identifying the unique circle.

Circle nomenclature

In the constructions below, a circle defined by a center point $P$ and a point on its circumference, $Q$, through which the arc of the circle passes, is denoted $P(Q)$. As most circles are not compass-drawn, center and circumference points are named explicitly. The arc, if drawn, may also be named, such as circle $c$. Per the theorem, when a compass-drawn circle is provided it is simply referred to as the given circle or the provided circle.

Circle generality

The provided circle should always be assumed to be placed arbitrarily in the plane with an arbitrary radius. Many examples of constructability with a straightedge one may find in various references on and offline, will presume that the circle is not placed in general position. Instead, for example, the constructability of a polygon may postulate that the circle is circumscribing. Such assumptions simplify a construction but does not prove generality of the claim of constructability. For the purposes of this theorem, we may assume that the circle is indeed fully general.

Usage of the arc of the provided circle(s)

The intersection points between any line and the given circle may be found directly, as can the intersection points between the arcs of two circles, if provided. The Poncelet-Steiner Theorem does not prohibit the normal treatment of circles already drawn in the plane; normal construction rules apply. The theorem only prohibits the construction of new circular arcs with a compass.

Arbitrary point placement

Steiner constructions and those constructions herein proving the Poncelet-Steiner theorem require the arbitrary placement of points in space. These constructions rely on the concept of fixed points, wherein the resultant construction is independent of the arbitrariness employed during construction. In some construction paradigms - such as in the geometric definition of the constructible number - arbitrary point placement may be prohibited. Traditional geometry has no such restriction on point placement; with such a restriction against the placement of arbitrary points, the single circle is indeed weaker than the compass. This can be reconciled, however. Steiner constructions may be used as a basis for the set of constructible numbers if one only enters into the set those points which are fixed, disregarding the arbitrarily placed points required during a construction.

Choice of construction among variants

In general constructions there are often several variations that will produce the same result. The choices made in such a variant can be made without loss of generality. However, when a construction is being used to prove that something can be done, it is not necessary to describe all these various choices and, for the sake of clarity of exposition, only one variant will be given below. The variants chosen below are done so for their ubiquity and generalizability in application rather than their simplicity or convenience under any particular set of special conditions.

Generality and simplicity

Some specific construction goals - such as for example the construction of a square - may potentially have relatively simple construction solutions, which will not be demonstrated here in the article, despite its simplicity. The omission of such constructions mitigate the length of the article. The purpose of these decisions is that such constructions may not be ubiquitous or sufficiently useful, particularly for the purposes of proving the theorem. Though the theorem and the constructions found herein can be used to construct any figure, no claims are made about the existence of simpler (straightedge-only) alternatives for any specific construction.

Alternative proofs

Alternative proofs do exist for the Poncelet-Steiner theorem, originating in an algebraic approach to geometry. Relying on equations and numerical values, this is a fairly modern interpretation which requires the notions of length, distance, and coordinate positions to be imported into the plane. This is well beyond the scope of traditional geometry. This article takes a more traditional approach and proves the theorem using pure geometric constructive techniques, which also showcases the practical application.

Constructive proof
The proof of the theorem and useful straightedge-only constructions follow.

Some preliminary constructions
To prove the above constructions #4 and #5, which are included below, a few necessary intermediary constructions are also explained below since they are used and referenced frequently. These are also straightedge-only constructions. All constructions below rely on basic constructions #1,#2,#3, and any other construction that is listed prior to it.

Parallel of a line having a colinear bisected segment
This construction does not require the use of the given circle. Naturally any line that passes through the center of the given circle implicitly has a bisected segment: the diameter is bisected by the center. The animated GIF file embedded at the introduction to this article demonstrates this construction - relying on the bisected diameter; the arc of the circle is never used - which is reiterated here without the circle and with enumerated steps.

Given an arbitrary line $n$ (in black) on which there exist two points $A$ and $B$, having a midpoint $M$ between them, and an arbitrary point $P$ in the plane (assumed not to be on line $n$) through which a parallel of line $n$ is to be made:
 * 1) Construct a line $\overline{AP}$ (in red).
 * 2) Construct a line $\overline{BP}$ (in orange).
 * 3) Define an arbitrary point $R$ on line $\overline{AP}$.
 * 4) Construct a line $\overline{BR}$ (in green).
 * 5) Construct a line $\overline{MR}$ (in light blue).
 * 6) Lines $\overline{MR}$ and $\overline{BP}$ intersect at point $X$.
 * 7) Construct a line $\overline{AX}$ (in magenta).
 * 8) Lines $\overline{BR}$ and $\overline{AX}$ intersect at point $Q$.
 * 9) Construct a line $\overline{PQ}$ (in dark blue), the desired parallel.

In some literature the bisected line segment is sometimes viewed as a one-dimensional "circle" existing on the line. Alternatively, some literature views the bisected line segment as a two dimensional circle in three dimensional space with the line passing through a diameter, but not parallel to the plane, thus intersecting the plane of construction at two points on the circumference with the midpoint simply being the prescribed circle center.

This construction is a special case of the projective harmonic conjugate construction, which is not demonstrated in this article.

Creating a bisected segment on a line
If the line passes through the center of a circle, the segment defined by the diameter through the circle is bisected by the center of the circle. In the general case, however, any other line in the plane may have a bisected segment constructed onto it. This construction does require the use of the given circle.

Given a line, $m$ (in black), and a circle centered at $A$, we wish to create points $E$, $B$, and $H$ on the line such that $B$ is the midpoint:
 * 1) Draw an arbitrary line (in red) passing through the given circles center, $A$, and the desired midpoint $B$ (chosen arbitrarily) on the line $m$.
 * 2) * Notice that the red line, $\overline{AB}$, passes through the center of the circle and highlights a diameter, bisected by the circle center. Any parallel may be made from this line according to the previous construction.
 * 3) Choose an arbitrary point $C$ on the given circle
 * 4) * For convenience the point should not lie on the perpendicular of line $\overline{AB}$ through the circle center.
 * 5) Construct a line (in orange), passing through $C$, that is parallel to the red line $\overline{AB}$.
 * 6) * If point $C$ is on the perpendicular of $\overline{AB}$ through the circle center, the parallel would be a tangent to the circle. The construction is possible by other means not listed in this article, but it is also unnecessary to choose such a point.
 * 7) * This parallel intersects the given circle at $D$.
 * 8) * This parallel also intersects the black line $m$ at $E$, defining one end of the line segment.
 * 9) Create two lines (in green), $\overline{AC}$ and $\overline{AD}$, that each pass through the given circles center.
 * 10) * These green lines intersect the given circle at points $G$ and $F$, respectively.
 * 11) Line $\overline{FG}$ (in blue) intersects the line $m$ at $H$, defining the other endpoint of the line segment.
 * 12) * Segment $\overline{EH}$ now exists coincident with line $m$ and having midpoint $B$.

As point $C$ is chosen arbitrarily, there is no need for it to inconveniently be on the perpendicular of line $\overline{AB}$ through the circle center. If however it is, line $\overline{CD}$ is merely the tangent line to the circle through point $C$, which is coincident to point $D$. This construction is possible though the construction is not listed in this article. Points $F$ and $G$ may be constructed as before, and will also equal one another. And again, line $\overline{GF}$ is merely the tangent line to the circle at that point. Thus points $E$, $H$ and their midpoint $B$ may be found, as before, with only a minor change adding a subconstruction.

Constructing a parallel of any line
This construction does require the use of the given circle. In order to generalize the parallel line construction to all possible lines, not just the ones with a collinear bisected line segment, it becomes necessary to have additional information. In keeping with the Poncelet-Steiner theorem, a circle (with center) is the object of choice for this construction.

To construct a parallel line of any given line, through any point in the plane, we trivially combine two constructions:
 * 1) Any line from which a parallel is to be made must have a bisected segment constructed onto it, if one does not already exist.
 * 2) A parallel is then constructed according to the previous parallel construction involving the collinear bisected segment.

In alternative constructions, which are not demonstrated in this article, a parallel may be constructed from any pair of lines which are already parallel to one another; thus a third parallel may be produced from any two, without the use of a circle. Additionally, a parallel of any line may be constructed whenever there exists in the plane any parallelogram, also without the use of a given circle.

Constructing a perpendicular line
This construction does require the use of the given circle and takes advantage of Thales's theorem.

From a given line $m$, and a given point $A$ in the plane, a perpendicular to the line is to be constructed through the point. Provided is the given circle $O(r)$.
 * 1) If the desired line from which a perpendicular is to be made, $m$, does not pass through the given circle - as is depicted - or it also passes through the given circle's center, then a new parallel line (in red) may be constructed arbitrarily such that it does pass through the given circle but not its center, and the perpendicular is to be made from this line instead.
 * 2) This red line which passes through the given circle but not its center, will intersect the given circle in two points, $B$ and $C$.
 * 3) Draw a line $\overline{BO}$ (in orange), through the circle center.
 * 4) * This line intersects the given circle at point $D$.
 * 5) Draw a line $\overline{DC}$ (in light green).
 * 6) * This line is perpendicular to the red (and therefore the black) lines, $\overline{BC}$ and $m$.
 * 7) Construct a parallel of line $\overline{DC}$ through point $A$ using previous constructions.
 * 8) * A perpendicular of the original black line, $m$, now exists in the plane, line $\overline{DC}$.
 * 9) * A parallel of any line may be constructed through any point in the plane.

If the line from which a perpendicular is to be made does pass through the circle center, an alternative approach would be to construct the tangent lines to the circle at the lines points of intersection, using Steiner constructions. This is not demonstrated in this article.

Another option in the event the line passes through the circles center would be to construct a parallel to it through the circle at an arbitrary point. An isosceles trapezoid (or potentially an isosceles triangle) is formed by the intersection points to the circle of both lines. The two non-parallel sides of which may be extended to an intersection point between them, and a line drawn from there through the circle's center. This line is perpendicular, and the diameter is bisected by the center.

By an alternative construction not demonstrated in this article, a perpendicular of any line may be constructed without a circle, provided there exists in the plane any square.

Constructing the midpoint of any segment (segment bisection)

Given is a line segment $\overline{AB}$, which is to be bisected. Optionally, a parallel line $m$ exists in the plane.

 * 1) If the line $m$, which is parallel to line segment $\overline{AB}$, does not exist in the plane then it must be constructed according to earlier constructions using the given circle in the plane (not depicted).
 * 2) * A given circle in the plane is not required for this construction if the parallel already exists.
 * 3) * The parallel may be placed in the plane arbitrarily, so long as it is not collinear with the line segment.
 * 4) Arbitrarily choose a point $C$ in the plane which is not collinear with the line or the line segment.
 * 5) * One may, alternatively, construct point $C$ as the intersection of lines $\overline{AD}$ and $\overline{BE}$, provided that points $D$ and $E$ already exist on a parallel line $m$.
 * 6) Draw a line $\overline{AC}$ (in red), intersecting line $m$ at point $D$.
 * 7) Draw a line $\overline{BC}$ (in orange), intersecting line $m$ at point $E$.
 * 8) Draw two lines, $\overline{AE}$ and $\overline{BD}$ (each in light green), intersecting each other at point $X$
 * 9) Draw a line $\overline{CX}$ (in blue), intersecting segment $\overline{AB}$ at point $M$.
 * 10) * Point $M$ is the desired midpoint of segment $\overline{AB}$.
 * 11) * Line $\overline{CX}$ also bisects segment $\overline{DE}$

For added perspective, in some sense this construction is a variant of a previous construction of a parallel from a bisected line segment, and is therefore also a special case of the projective harmonic conjugate (not provided in this article). It is the same set of lines when taken on whole, but constructed in a different order, and from a different initial set of conditions, arriving at a different end goal.

Since any arbitrary segment on one of two parallel lines can be bisected, and any line with a bisected segment on it may have a parallel constructed, the two scenarios are geometrically equivalent propositions. They imply one another; a simple construction can convert one scenario into the other using no additional information.

In an alternative construction, any line segment may be bisected whenever a parallelogram exists in the plane.

Rotating a line segment
To define a circle only the center and one point - any point - on the circumference is required. In principle a new point $B' $ is constructed such that circle $A(B)$ is equal to circle $A(B')$, though the point $B$ is not equal to point $B' $. In essence, segment $\overline{AB}$ is rotated about the axis point $A$, to $\overline{AB' }$, for a different set of defining points for the same circle.

One way of going about this which satisfies most conditions is as follows:
 * 1) Draw the line segment $\overline{AB}$ (in black).
 * 2) Construct a parallel (in red) of line $\overline{AB}$ through the center, point $O$, of the given circle.
 * 3) * The parallel intersects the given circle at some point $b$.
 * 4) Choose a point $b'$ arbitrarily on the given circle not colinear with line $\overline{Ob}$.
 * 5) Draw a line $\overline{Ob' }$ (in orange).
 * 6) Construct a parallel line to $\overline{Ob' }$ through point $A$ (in magenta).
 * 7) Draw a line $\overline{bb' }$ (in light green), connecting the points on the circles circumference.
 * 8) Construct a parallel of line $\overline{bb' }$ through point $B$ (in blue).
 * 9) Intersect the blue and pink parallels, from points $B$ and $A$, respectively.
 * 10) * This is point $B'$.
 * 11) * Point $B' $ is the desired point, rotating the line segment and defining the same circle centered at $A$.

This construction will fail if the desired rotation is diametrically opposite the circle (i.e. a half-circle rotation). One solution to this scenario is to employ two separate rotation constructions, neither one a half-circle rotation from the previous, one acting as an intermediary step. Choose any rotation angle arbitrarily, complete the rotation, then choose the supplementary angle and perform the rotation a second time.

There does exist a second, alternative rotation construction solution, based on projections and perspective points. Though it avoids the aforementioned half-circle rotation complication, it does have its own complications, which are similarly resolved with intermediary rotation constructions. The construction is no more versatile. It is not demonstrated in this article.

Constructing the radical axis between circles
This construction does require the use of the given circle (which is not depicted) for the referenced sub-constructions previously demonstrated.

Suppose two circles $A(B)$ and $C(D)$ are implicitly given, defined only by the points $A$, $B$, $C$, and $D$ in the plane, with their centers defined, but are not compass-constructed. The radical axis, line $m$ (in dark blue), between the two circles may be constructed:
 * 1) Draw a line $\overline{AC}$ (in orange) through the circle centers.
 * 2) Draw a line segment $\overline{BD}$ (in red) between the points on the circumference of the circles.
 * 3) Find the midpoint, $M$, of segment $\overline{BD}$.
 * 4) Draw lines $\overline{AM}$ and $\overline{CM}$ (both in light green), connecting the segment midpoint with each of the circle centers.
 * 5) Construct a line $j$ (in purple) passing through point $B$, and perpendicular to $\overline{AM}$.
 * 6) * Line $j$ is the radical axis between circle $M(B)$ and circle $A(B)$.
 * 7) Construct a line $k$ (in dark green) passing through point $D$, and perpendicular to $\overline{CM}$.
 * 8) * Line $k$ is the radical axis between circle $M(B)$ and circle $C(D)$.
 * 9) Lines $j$ and $k$ intersect at point $X$.
 * 10) * If the lines $j$ and $k$ are parallel then the segment midpoint $M$ is on the line $\overline{AC}$ - the centers of circles $A(B)$, $C(D)$, and $M(B)$ are colinear -  and the construction will fail. An alternative approach is required (see below).
 * 11) * Point $X$ is the power center between circles $A(B)$, $C(D)$, and $M(B)$, and therefore is the unique point that lies on the radical axis between any two of the three circles.
 * 12) * By transitivity, therefore, point $X$ exists on the radical axis between circles $A(B)$ and $C(D)$.
 * 13) Construct a line $m$ (in dark blue) perpendicular to line $\overline{AC}$ and passing through point $X$.
 * 14) * Line $m$ is the desired radical axis.

In the event that the construction of the radical axis fails due to there not being an intersection point $X$ between parallel lines $j$ and $k$, which results from the coincidental placement of the midpoint $M$ on the line $\overline{AC}$, an alternative approach is required. One such approach is to rotate the segment $\overline{AB}$ about the axis point $A$ (the center of circle $A(B)$). Once arrived at the arbitrary rotation $\overline{AB' }$, which defines the same circle, the radical axis construction can begin anew without issue.

Intersecting a line with a circle (Construction #4)
This construction does require the use of the provided circle, $O(r)$. Any line may be naturally intersected with any compass-drawn circle.

Given is the line $m$ (in black) and the circle $P(Q)$, which is not compass-constructed. The intersection points of the circle $P(Q)$ and the line $m$, which are point $A$ and $B$, may be constructed:
 * 1) Draw a line $\overline{PQ}$ (in red) through the points defining the circle.
 * 2) * If point $O$ is colinear with line $\overline{PQ}$ then segment $\overline{PQ}$ will need to be rotated about the center point $P$ of the circle, and this construction restarted anew.
 * 3) Construct a parallel (in orange) of line $\overline{PQ}$ through the center $O$ of the provided circle.
 * 4) * The parallel intersects the provided circle at two points, one of which is arbitrarily chosen: $R$.
 * 5) Draw a line $\overline{PO}$ (in light green), through the centers of the two circles (i.e. the one provided by compass construction and the one which is to be intersected).
 * 6) Draw a line $\overline{QR}$ (in light blue), connecting the two points on the circumferences of the two circles.
 * 7) Intersect the lines $\overline{PO}$ and $\overline{QR}$ at point $X$.
 * 8) * If point $X$ does not exist due to lines $\overline{PO}$ and $\overline{QR}$ being parallel - which results from circles $P(Q)$ and $O(r)$ having equal radii - then refer back to step 2 and choose the alternate point of intersection, $R$.
 * 9) Choosing a point $M$ arbitrarily on line $m$, such that it is not on line $\overline{PO}$, draw a line $\overline{PM}$ (in magenta).
 * 10) * For construction simplicity and only if line $\overline{PQ}$ is not parallel to line $m$, lines $\overline{PM}$ and $\overline{PQ}$ may be coincident.
 * 11) Draw a line $\overline{MX}$ (in brown).
 * 12) Construct a parallel (in dark purple) of line $\overline{PM}$ through the center $O$ of the provided circle.
 * 13) * The parallel intersects the line $\overline{MX}$ at a point $N$.
 * 14) Construct a parallel (in yellow) of line $m$ through the point $N$.
 * 15) * The parallel intersects the provided circle at points $C$ and $D$.
 * 16) * If the parallel does not intersect the provided circle then neither does the line $m$ intersect circle $P(Q)$.
 * 17) Draw lines $\overline{CX}$ and $\overline{DX}$ (both in dark blue).
 * 18) * These lines both intersect line $m$ at points $A$ and $B$, respectively.
 * 19) Points $A$ and $B$ are the desired points of intersection between the line $m$ and the circle $P(Q)$.

Intersecting two circles (Construction #5)
The intersection between two circles becomes a trivial combination of two earlier constructions:
 * 1) Construct the radical axis between the two circles.
 * 2) Construct the intersection points between the radical axis and either one of the two circles arbitrarily chosen:
 * 3) * The radical axis is a line, so this construction is possible per the previous construction: basic construction #4.
 * 4) These points are the desired points of intersection of the circles.
 * 5) * The two circles and the radical axis all intersect at the same loci of points: two points, one point if tangential, or none if they do not intersect.
 * 6) * If the radical axis does not intersect one circle then it intersects neither, and neither do the two circles intersect.

A circle through one point centered at another point (Construction #2, revisited)
The second basic construction - describing a full circle with just two defining points - never needed an arc to be constructed with the compass in order for the circle to be utilized in constructions. Namely, the intersections of circles both with circles and with lines, which together are the essence of all constructions involving a circle, are achievable without the arc. Thus defining a circle by its center and by any arbitrary point on its circumference is sufficient to fully describe the entire circle and construct with it. As such, the arc only serves an aesthetic purpose. Basic construction #2 is satisfied.

Conclusion
Since all five basic constructions have been shown to be achievable with only a straightedge, provided that a single circle with its center is placed in the plane, this proves the Poncelet-Steiner theorem.

Projective Geometry and the Poncelet-Steiner Theorem
Though a distinct topic in its own right, many of the concepts of projective geometry are applied here to Steiner constructions. Jean-Victor Poncelet was a major contributor to the subject when he postulated the theorem of this article, which Jakob Steiner later proved. Many of the related concepts developed in projective geometry include but are not limited to: concurrence, "points at infinity", perspective, ratios and cross ratios, stable or fixed points of involutions, invariants, homogeneity, linear transformations, projective harmonics, and others. A thorough treatment of Steiner constructions and their proofs require a background in projective geometry, though the subject of projective geometry is not restricted to straightedge-only constructions.

Restricted constructions involving the compass
The Poncelet–Steiner theorem can be contrasted with the Mohr–Mascheroni theorem, which states that any compass and straightedge construction can be performed with only a compass. The straightedge is not required but for aesthetic purposes; nothing else is needed in the plane.

The rusty compass restriction allows the use of a compass and straightedge, provided that the compass produces circles of fixed radius. Although the rusty compass constructions were explored since the 10th century, and all of Euclid was shown to be constructable with a rusty compass by the 17th century, the Poncelet-Steiner theorem proves that the rusty compass and straightedge together are more than sufficient for any and all Euclidean construction. Indeed, the rusty compass becomes a tool simplifying constructions over merely the straightedge and single circle. Viewed the other way, the Poncelet-Steiner theorem not only fixes the width of the rusty compass, but ensures that the compass breaks after its first use.

The compass equivalence theorem proves that the rigid compass (also called the modern compass) - one that holds its spacing when lifted from the plane - is equivalent to the traditional collapsing compass (also called divider) - one that does not retain its spacing, thus "resetting to zero", every time it is lifted from the plane. The ability to transfer distances (i.e. construct congruent circles) - an operation made trivial by the rigid compass - was proven by Euclid to be possible with the collapsing compass. In fact it can be done using only the collapsing compass, without the straightedge tool. Consequently the rigid compass and collapsing compasses are equivalent; what can be constructed by one can be constructed by the other, even in the compass-only construction paradigm.

Restricted Steiner constructions
The requirement placed on the Poncelet-Steiner theorem - that one circle with its center provided exist in the plane - has been since generalized, or strengthened, to include alternative but equally restrictive conditions.

Other unique scenarios undoubtedly exist than those listed here. This is not an exhaustive list of possibilities.

Poncelet-Steiner without the circle center
In two such alternatives, the centre may be omitted entirely provided that given are either two concentric circles, or two distinct intersecting circles, of which there are two cases: two intersection points and one intersection point (tangential circles). From any of these scenarios, centres can be constructed, reducing the scenario to the original hypothesis. These do not contradict Steiner's theorem which, although stating a center is absolutely required, also hypothesizes only one circle exists.

Still other variations exist. It suffices to have two non-intersecting, non-concentric circles (without their centres), provided that at least one point is given on either the centerline through them or on the radical axis between them, or provided any two parallel lines arbitrarily in the plane. It also suffices, alternatively, to have three non-intersecting circles. Once a single center is constructed, the scenario again reduces to the original hypothesis of the Poncelet-Steiner theorem.

In each of these scenarios, more information exists than just a second circle. The fact that two circles are concentric, or that two circles have known intersection points, or the presence of a third circle, or of a point on either a centerline or a radical axis, constitutes an additional piece of information beyond merely the presence of a second circle.

Poncelet-Steiner without the full circular arc
In another alternative, the entire circle is not required at all. In 1904, Francesco Severi proved that any small arc (of the circle), together with the centre, will suffice. This construction breaks the rusty compass at any point before the first circle is completed, but after it has begun, thus drawing some continuous portion of the arc of the circle in the plane, and still all constructions remain possible. Thus, the conditions hypothesizing the Poncelet-Steiner theorem may indeed be weakened, but only with respect to the completeness of the circular arc, and not, per the Steiner theorem, with respect to the center.

The theorem demonstrates the Steiner construction of the intersection points between a line and the circle of an arc, regardless of the size or position of the arc, using only a straightedge and the arc. The construction, also, does not make use of the center of the circle of the arc. Though the center is required to complete all of Euclid, as the Steiner theorem and the Poncelet-Steiner theorem both proves, the center is not needed in order to intersect a line with the circle of that arc. Using this construction, the arc, and the center of the circle of the arc, all of the above Poncelet-Steiner constructions are equally achievable, albeit with greater difficulty.

Also in each of the previously mentioned two-or-more-circle scenarios wherein the circle centers are omitted, the completeness of the circular arc is not necessary, as per Severi's theorem, provided that circle intersections are explicitly given whenever arcs of the circle(s) do not exist where intersection points do. That is, if the intersection points between two circles of arcs cannot be found directly by way of the two arcs, they must be provided. The completeness of the circular arc is otherwise redundant.

Extended, liberated, or neusis constructions
Instead of restricting the rules of construction further, it is of equal interest to study relaxing the restrictions. These are sometimes called extended constructions, because they extend what is constructible by extending the allowable toolset. These constructions are also called neusis constructions (from Greek) because they employ tools other than the compass and straightedge, or liberated constructions because they alleviate the restrictions of the traditional paradigm.

Just as geometers have studied what remains possible to construct (and how) when additional restrictions are placed on traditional construction rules - such as compass only, straightedge only, rusty compass, etc. - they have also studied what constructions becomes possible that weren't already when the natural restrictions inherent to traditional construction rules are alleviated. Questions such as "what becomes constructible", "how might it be constructed", "what are the fewest traditional rules to be broken", "what are the simplest tools needed", "which seemingly different tools are equivalent", "how does the new paradigm simplify traditional constructions", etc. are asked.

The arbitrary angle is not trisectable using traditional compass and straightedge rules, for example, but the trisection becomes constructible when allowed the additional tool of an ellipse in the plane, which is itself not constructible. Some of the traditional unsolved problems such as angle trisection, doubling the cube, squaring the circle, finding cubic roots, etc., since proven to be impossible by straightedge and compass alone, have been resolved using an expanded set of tools. In general, the objects studied to extend the scope of what is constructible have included:
 * Non-constructible "auxiliary" curves in the plane - including any of the conic sections, cycloids, lemniscates, limaçons, the Archimedean spiral, Euler spiral, any of the trisectrices or quadratrices, and others.
 * Physical tools other than the compass and straightedge - generally called neuseis (plural) - which include specific tools such as the Tomahawk, markable and double-edged straightedges, graduated rulers, right triangular rulers, linkages, ellipsographs, French curves, and others.
 * Paper-folding techniques, derived from the mathematical and geometric study of Origami.

Each of the three above categorical approaches have their own unique angle trisection solutions.

The ancient geometers considered the compass and straightedge constructions (known as planar constructions) as ideal and preferred. Second to that they preferred solid constructions, which included the use of conic sections in the plane other than the circle. They favored thirdly the use of arbitrary smooth curves in the plane (such as the Archimedean spiral), and least of all the use of neuseis (alternative physical handheld tools). It is doubtful that the ancient geometers - at least of the western world - even considered paper folding.

The term neusis (singular) or neusis construction may also refer to a specific tool or method employed by the ancient geometers.

The graduated ruler is unique in that it defines a metric, and also a norm, and gives rise to the algebraic treatment of geometry, Cartesian graphing, and imports a standard unit for proportionality of segments. The algebraic approach to geometry offers a distinct proof of the Poncelet-Steiner theorem.

Approximations
It is worthwhile to point out that in all construction paradigms, the implicit rule is that all constructions must terminate in a finite number of applications of the available tools (compass and straightedge), and produce the exact intended results. Entire discussions could be made with either of these conditions alleviated.

For any otherwise non-constructible figure:
 * It is possible to approximate a construction to a predetermined level of accuracy using only compass and straightedge, using a reiterative approach.
 * Exact construction may become possible at the infinite limit of this convergent process, but requires infinite constructions.

For example, an angle trisection may be performed exactly with compass and straightedge, using an infinite sequence of angle bisections. If the construction is terminated at some finite iteration, an accurate approximation of a trisection can be achieved to arbitrary accuracy. Although each point, line or circle is a valid construction, what it aims to approximate can never truly be achieved in finite applications of a compass and/or straightedge.

There are, alternatively, exactly constructible non-iteratively constructed figures that are reasonable approximations for non-constructible figures. For example, there is a relatively simple non-iterative constructions for an approximation of the heptagon.

Higher dimensional generalizations
The Poncelet-Steiner theorem has been generalized to higher dimensions, such as, for example, a three dimensional variation where the straightedge is replaced with a plane, and the circle with center is replaced with a sphere with center. This is essentially a "straightedge only" variation of three dimensional geometry. Though research is ongoing, and some propositions are yet to be proved, many of the properties that apply to the two dimensional case also apply to higher dimensions, as implementations of projective geometry.

title: Poncelet–Steiner theorem


text:
Waterdeep (disambiguation)

Waterdeep is a fictional city in the Forgotten Realms campaign setting for the Dungeons & Dragons role-playing game.

Waterdeep may also refer to:
 * Waterdeep (band), a Christian band
 * Waterdeep (module), a D&D game adventure taking place within the fictional city
 * Waterdeep (novel), a D&D novel in the Avatar Series
 * City of Splendors: Waterdeep, a supplement to the D&D game

title: Waterdeep (disambiguation)