**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.

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] }

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

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] }

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] }

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] }

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.

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 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 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, an **equilateral triangle** is a triangle in which all three sides have the same length. 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**.

In geometry, **Thales's theorem** states that if A, B, and C are distinct points on a circle where the line AC is a diameter, the angle ABC is a right angle. Thales's 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 but it is sometimes attributed to Pythagoras.

In geometry, **symmedians** are three particular geometrical lines associated with every triangle. They are constructed by taking a median of the triangle, and reflecting the line over the corresponding angle bisector. The angle formed by the **symmedian** and the angle bisector has the same measure as the angle between the median and the angle bisector, but it is on the other side of the angle bisector.

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, **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.

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.

**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. It is a theorem commonly employed in various math competitions, secondary school mathematics examinations, and has wide applicability to many problems in the real world.

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 Euclidean geometry, a **right kite** is a kite that can be inscribed in a circle. That is, it is a kite with a circumcircle. Thus the right kite is a convex quadrilateral and has two opposite right angles. If there are exactly two right angles, each must be between sides of different lengths. All right kites are bicentric quadrilaterals, since all kites have an incircle. One of the diagonals divides the right kite into two right triangles and is also a diameter of the circumcircle.

**Pappus's area theorem** describes the relationship between the areas of three parallelograms attached to three sides of an arbitrary triangle. The theorem, which can also be thought of as a generalization of the Pythagorean theorem, is named after the Greek mathematician Pappus of Alexandria, who discovered it.

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

**William Chapple** (1718–1781) was an English surveyor and mathematician. His mathematical discoveries were mostly in plane geometry and include:

- ↑ 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. JSTOR 2695499.

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