Paul Yiu is currently at Florida Atlantic University. In his Notes on Euclidean Geometry (1998), he poses the following interesting problem on p. 2: If in a square ABCD, equilateral triangles AHD and CDI are both constructed towards the inside or outside, and rays BI and BH respectively meet sides AD and CD in P and Q, then triangle BPQ is equilateral.
Drag any of the red points (vertices) to dynamically move and change the position, orientation or size of the square.
Paul Yiu's Problem
1) Can you explain why (prove that) the result is true when the equilateral triangles are consructed towards the inside as above? If so, can you extend the argument to cover the case when the equilateral triangles are constructed towards the outside? Only after spending some time with it and you're still stuck with it, go to Hint.
2) Can you generalize the result to another quadrilateral? Think about it a bit before going to A generalization of Paul Yiu's problem.
Michael de Villiers, 5 June 2011.