Thébault's theorem
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.
Thébault's problem I
Given any
It is a special case of van Aubel's theorem and a square version of the Napoleon's theorem.
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.[2]
Thébault's problem III
Given any
Until 2003, academia thought this third problem of Thébault the most difficult to
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.
References
- ^ 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
- MR 2055379
- JSTOR 2695499.
External links
- Thébault's problems and variations at cut-the.knot.org