Paul Yiu's theorem can be generalized to a rhombus as follows: If in a rhombus 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 extended in P and Q, then triangle BPQ is equilateral.
Drag any of the red points (vertices) A, B or C to dynamically move and change the position, orientation or size of the square.
A generalization of Paul Yiu's theorem
Challenge
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.
Additional Property
Click on the 'Show Further Property' button in the sketch above. What do you notice about the area result now?
Further generalization
Now try the same construction for a parallelogram. What do you notice? Can you also prove it?
Still Further generalization
Thanos Kalogerakis from Corinth, Greece, neatly generalized the result (April 2021) in the 'Romantics of Geometry (Ρομαντικοί της Γεωμετρίας)' Group on Facebook to any quadrilateral as shown in the following sketch. Can you also prove this result?
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Michael de Villiers, 5 June 2011; updated 7 & 10 April 2021.