Paul Yiu was at Florida Atlantic University, and published many papers on Euclidean Geometry, but is apparently now retired. He was the Founding Editor of the online geometry research journal Forum Geometricorum.

In his very useful *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) *B* or *C* to dynamically move and change the position, orientation or size of the square.

Paul Yiu's Theorem

**Challenge**

1) Can you *explain why* (prove that) the result is true when the equilateral triangles are constructed 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 theorem.

**Additional Property**

Click on the '**Show Further Property**' button in the sketch above. What do you notice? Can you explain why (prove that) it is true?

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Michael de Villiers, 5 June 2011; updated 7 April 2021.