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**Thébault's theorem** is the name given variously to one of the geometry problems proposed by the French mathematician Victor Thébault, individually known as Thébault's problem I, II, and III.

**Geometry** is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer.

**France**, officially the **French Republic**, is a country whose territory consists of metropolitan France in Western Europe and several overseas regions and territories. The metropolitan area of France extends from the Mediterranean Sea to the English Channel and the North Sea, and from the Rhine to the Atlantic Ocean. It is bordered by Belgium, Luxembourg and Germany to the northeast, Switzerland and Italy to the east, and Andorra and Spain to the south. The overseas territories include French Guiana in South America and several islands in the Atlantic, Pacific and Indian oceans. The country's 18 integral regions span a combined area of 643,801 square kilometres (248,573 sq mi) and a total population of 67.3 million. France, a sovereign state, is a unitary semi-presidential republic with its capital in Paris, the country's largest city and main cultural and commercial centre. Other major urban areas include Lyon, Marseille, Toulouse, Bordeaux, Lille and Nice.

A **mathematician** is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems.

Given any parallelogram, construct on its sides four squares external to the parallelogram. The quadrilateral formed by joining the centers of those four squares is a square.^{ [1] }

In Euclidean geometry, a **parallelogram** is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations.

In Euclidean plane geometry, a **quadrilateral** is a polygon with four edges and four vertices or corners. Sometimes, the term **quadrangle** is used, by analogy with triangle, and sometimes **tetragon** for consistency with pentagon (5-sided), hexagon (6-sided) and so on.

It is a special case of van Aubel's theorem and a square version of the Napoleon's theorem.

In plane geometry, **Van Aubel's theorem** describes a relationship between squares constructed on the sides of a quadrilateral. Starting with a given quadrilateral, construct a square on each side. Van Aubel's theorem states that the two line segments between the centers of opposite squares are of equal lengths and are at right angles to one another. Another way of saying the same thing is that the center points of the four squares form the vertices of an equidiagonal orthodiagonal quadrilateral. The theorem is named after H. H. van Aubel, who published it in 1878.

In geometry, **Napoleon's theorem** states that if equilateral triangles are constructed on the sides of any triangle, either all outward or all inward, the lines connecting the centres of those equilateral triangles themselves form an equilateral triangle.

Given a square, construct equilateral triangles on two adjacent edges, either both inside or both outside the square. Then the triangle formed by joining the vertex of the square distant from both triangles and the vertices of the triangles distant from the square is equilateral.^{ [2] }

In geometry, an **equilateral triangle** is a triangle in which all three sides are equal. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. It is also a regular polygon, so it is also referred to as a **regular triangle**.

Given any triangle ABC, and any point M on BC, construct the incircle and circumcircle of the triangle. Then construct two additional circles, each tangent to AM, BC, and to the circumcircle. Then their centers and the center of the incircle are colinear.^{ [3] }^{ [4] }

A **triangle** is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices *A*, *B*, and *C* is denoted .

In geometry, the **tangent line** to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve *y* = *f* (*x*) at a point *x* = *c* on the curve if the line passes through the point (*c*, *f* ) on the curve and has slope *f*'(*c*) where *f*' is the derivative of *f*. A similar definition applies to space curves and curves in *n*-dimensional Euclidean space.

Until 2003, academia thought this third problem of Thébault the most difficult to prove. It was published in the American Mathematical Monthly in 1938, and proved by Dutch mathematician H. Streefkerk in 1973. However, in 2003, Jean-Louis Ayme discovered that Y. Sawayama, an instructor at The Central Military School of Tokyo, independently proposed and solved this problem in 1905.^{ [5] }

In mathematics, a **proof** is an inferential argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms, along with accepted rules of inference. Axioms may be treated as conditions that must be met before the statement applies. Proofs are examples of exhaustive deductive reasoning or inductive reasoning and are distinguished from empirical arguments or non-exhaustive inductive reasoning. A proof must demonstrate that a statement is always true, rather than enumerate many confirmatory cases. An unproved proposition that is believed to be true is known as a conjecture.

* The American Mathematical Monthly* is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America.

The **Netherlands** is a country located mainly in Northwestern Europe. The European portion of the Netherlands consists of twelve separate provinces that border Germany to the east, Belgium to the south, and the North Sea to the northwest, with maritime borders in the North Sea with Belgium, Germany and the United Kingdom. Including three island territories in the Caribbean Sea—Bonaire, Sint Eustatius and Saba— it forms a constituent country of the Kingdom of the Netherlands. The official language is Dutch, but a secondary official language in the province of Friesland is West Frisian.

An "external" version of this theorem, where the incircle is replaced by an excircle and the two additional circles are external to the circumcircle, is found in Shay Gueron (2002). ^{ [6] } A proof based on Casey's theorem is in the paper.

**Area** is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analog of the length of a curve or the volume of a solid.

In geometry, the **nine-point circle** is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are:

In geometry, the **incircle** or **inscribed circle** of a triangle is the largest circle contained in the triangle; it touches the three sides. The center of the incircle is a triangle center called the triangle's incenter.

In geometry, three or more than three straight lines make a polygon and an **equilateral polygon** is a polygon which has all sides of the same length. Except in the triangle case, it need not be equiangular, but if it does then it is a regular polygon. If the number of sides is at least five, an equilateral polygon need not be a convex polygon: it could be concave or even self-intersecting.

In geometry, **Thales' theorem** states that if A, B, and C are distinct points on a circle where the line AC is a diameter, then the angle ∠ABC is a right angle. Thales' theorem is a special case of the inscribed angle theorem, and is mentioned and proved as part of the 31st proposition, in the third book of Euclid's *Elements*. It is generally attributed to Thales of Miletus, who is said to have offered an ox as a sacrifice of thanksgiving for the discovery, but sometimes it is attributed to Pythagoras.

In geometry, the **Japanese theorem** states that no matter how one triangulates a cyclic polygon, the sum of inradii of triangles is constant.

In geometry, given a triangle *ABC* and a point *P* on its circumcircle, the three closest points to *P* on lines *AB*, *AC*, and *BC* are collinear. The line through these points is the **Simson line** of *P*, named for Robert Simson. The concept was first published, however, by William Wallace in 1799.

In geometry, an **inscribed** planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figure G" means precisely the same thing as "figure G is circumscribed about figure F". A circle or ellipse inscribed in a convex polygon is tangent to every side or face of the outer figure. A polygon inscribed in a circle, ellipse, or polygon has each vertex on the outer figure; if the outer figure is a polygon or polyhedron, there must be a vertex of the inscribed polygon or polyhedron on each side of the outer figure. An inscribed figure is not necessarily unique in orientation; this can easily be seen, for example, when the given outer figure is a circle, in which case a rotation of an inscribed figure gives another inscribed figure that is congruent to the original one.

In geometry, a **centre** of an object is a point in some sense in the middle of the object. According to the specific definition of centre taken into consideration, an object might have no centre. If geometry is regarded as the study of isometry groups then a centre is a fixed point of all the isometries which move the object onto itself.

**Pompeiu's theorem** is a result of plane geometry, discovered by the Romanian mathematician Dimitrie Pompeiu. The theorem is simple, but not classical. It states the following:

**Viviani's theorem**, named after Vincenzo Viviani, states that the sum of the distances from *any* interior point to the sides of an equilateral triangle equals the length of the triangle's altitude.

In geometry, the **Japanese theorem** states that the centers of the incircles of certain triangles inside a cyclic quadrilateral are vertices of a rectangle.

In geometry, **Napoleon points** are a pair of special points associated with a plane triangle. It is generally believed that the existence of these points was discovered by Napoleon Bonaparte, the Emperor of the French from 1804 to 1815, but many have questioned this belief. The Napoleon points are triangle centers and they are listed as the points X(17) and X(18) in Clark Kimberling's Encyclopedia of Triangle Centers.

In Euclidean geometry, the **trillium theorem** – is a statement about properties of inscribed and circumscribed circles and their relations.

**Van Schooten's theorem**, named after the Dutch mathematician Frans van Schooten, describes a property of equilateral triangles. It states:

- ↑ http://www.cut-the-knot.org/Curriculum/Geometry/Thebault1.shtml (retrieved 2016-01-27)
- ↑ http://www.cut-the-knot.org/Curriculum/Geometry/Thebault2.shtml (retrieved 2016-01-27)
- ↑ http://www.cut-the-knot.org/Curriculum/Geometry/Thebault3.shtml (retrieved 2016-01-27)
- ↑ Alexander Ostermann, Gerhard Wanner:
*Geometry by Its History*. Springer, 2012, pp. 226–230 - ↑ Ayme, Jean-Louis (2003), "Sawayama and Thébault's theorem" (PDF),
*Forum Geometricorum*,**3**: 225–229, MR 2055379 - ↑ Gueron, Shay (April 2002). "Two Applications of the Generalized Ptolemy Theorem" (PDF).
*The American Mathematical Monthly*.**109**(4): 362–370. doi:10.2307/2695499.

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- Thébault's problems and variations at cut-the.knot.org

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