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

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

problems proposed by the French

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

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

Thébault's problem I

Given any parallelogram, construct on its sides four

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

s external to the parallelogram. The

quadrilateral In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...

formed by joining the centers of those four squares is a square. It is a special case of van Aubel's theorem and a square version of the

Napoleon's theorem 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 tr ...

Thébault's problem II

Given a square, construct

equilateral triangle 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 oth ...

s 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 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 \triangle ABC. In Euclidean geometry, any three points, when non- colline ...

ABC, and any point M on BC, construct the

incircle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...

and

circumcircle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

of the triangle. Then construct two additional circles, each

tangent In geometry, the tangent line (or simply tangent) 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. Mo ...

to AM, BC, and to the circumcircle. Then their centers and the center of the incircle are colinear. 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 Dutch commonly refers to: * Something of, from, or related to the Netherlands * Dutch people () * Dutch language () Dutch may also refer to: Places * Dutch, West Virginia, a community in the United States * Pennsylvania Dutch Country People E ...

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. 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). A proof based on

Casey's theorem In mathematics, Casey's theorem, also known as the generalized Ptolemy's theorem, is a theorem in Euclidean geometry named after the Irish mathematician John Casey. Formulation of the theorem Let \,O be a circle of radius \,R. Let \,O_1, O_2 ...

is in the paper.

References

External links

Thébault's problems and variations
at cut-the.knot.org {{DEFAULTSORT:Thebault's theorem Theorems about quadrilaterals Theorems about triangles and circles