**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, who is said to have offered an ox, probably to the god Apollo, as a sacrifice of thanksgiving for the discovery, 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, 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 center taken into consideration, an object might have no center. If geometry is regarded as the study of isometry groups then a center 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. 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 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.

**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 include the first proof of the existence of the orthocentre of a triangle, a formula for the distance between the incentre and circumcentre of a triangle, and the discovery of Poncelet's porism on triangles with a common incircle and circumcircle. He was also one of the earliest mathematicians to calculate the values of annuities.

- ↑ 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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