Thébault's theorem is the name given variously to one of the geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

 problems proposed by the French

France

The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...

 mathematician

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

 Victor Thébault

Victor Thébault

Victor Thébault was a French mathematician best known for propounding three problems in geometry. Thébault's theorem is used in some references to refer to the first of these problems, in other references to the third....

, individually known as Thébault's problem I, II, and III.

Thébault's problem I

Given any parallelogram

Parallelogram

In Euclidean geometry, a parallelogram is a convex 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...

, construct on its sides four square

Square (geometry)

In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...

s external to the parallelogram. The quadrilateral

Quadrilateral

In Euclidean plane geometry, a quadrilateral is a polygon with four sides and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on...

 formed by joining the centers of those four squares is a square.

It is a special case of van Aubel's theorem

Van Aubel's theorem

In 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 holds that the two line segments connecting the centers of opposite squares are of equal...

.

Thébault's problem II

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.

Thébault's problem III

Given any triangle

Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

 ABC, and any point M on BC, construct the incircle and circumcircle of the triangle. Then construct two additional circles, each tangent

Tangent

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. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...

 to AM, BC, and to the circumcircle. Then their centers and the center of the incircle are colinear.

Until 2003, acadamia thought this third problem of Thébault the most difficult to prove

Mathematical proof

In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...

. It was published in the American Mathematical Monthly

American Mathematical Monthly

The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894. It is currently published 10 times each year by the Mathematical Association of America....

 in 1938, and proved by Dutch

Netherlands

The Netherlands is a constituent country of the Kingdom of the Netherlands, located mainly in North-West Europe and with several islands in the Caribbean. Mainland Netherlands borders the North Sea to the north and west, Belgium to the south, and Germany to the east, and shares maritime borders...

 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.

External links

• Thébault's Problem I