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

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

problems proposed by the

French French may refer to: * Something of, from, or related to France ** French language, which originated in France ** French people, a nation and ethnic group ** French cuisine, cooking traditions and practices Arts and media * The French (band), ...

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, mathematical structure, structure, space, Mathematica ...

Victor Thébault Victor Michael Jean-Marie Thébault (1882–1960) was a French mathematician best known for propounding three problems in geometry. The name Thébault's theorem is used by some authors to refer to the first of these problems and by others to refe ...

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

Thébault's problem I

Given any

parallelogram In Euclidean geometry, a parallelogram is a simple polygon, simple (non-list of self-intersecting polygons, self-intersecting) quadrilateral with two pairs of Parallel (geometry), parallel sides. The opposite or facing sides of a parallelogram a ...

, construct on its sides four

square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

s external to the parallelogram. The

quadrilateral In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''l ...

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

van Aubel'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 convex quadrilateral, construct a square (geometry), square, external to the quadrilateral, on each ...

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 ...

. All three of these theorems are just a special case of Petr–Douglas–Neumann theorem. Thebault problem I tiling pattern

Thébault's problem II

Given a square, construct

equilateral triangle An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...

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 corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

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 triangle is a circle that passes through all three vertex (geometry), vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumrad ...

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, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...

to AM, BC, and to the circumcircle. Then their centers and the center of the incircle are collinear. Until 2003, academia thought this third problem of Thébault the most difficult to prove. It was published in the

American Mathematical Monthly ''The American Mathematical Monthly'' is a peer-reviewed scientific journal of mathematics. It was established by Benjamin Finkel in 1894 and is published by Taylor & Francis on behalf of the Mathematical Association of America. It is an exposi ...

in 1938, and proved by

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mathematician H. Streefkerk in 1973. However, in 2003, Jean-Louis Ayme discovered that , 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 Note that in the degenerate case, where all four ci ...

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