# Thread: A nice, unsolved geometry problem

1. ## A nice, unsolved geometry problem

The following geometry question looks simple but still unsolved.
Fox 223 (I am walking 8 foxes at the top of my head and trying to keep their tails apart !)
Creative Unusual Geometry Problems: Fox 223

This also looks related to 223:
Creative Unusual Geometry Problems: Fox 224

2. There is a theorem, (easily proved using similar triangles), involving the intersection of two chords in a circle.
If as a result of the intersection, one chord splits into lengths a and b, while the other splits into lengths p and q, then ab = pq.
Apply this at each of the four corners of the rhombus, add the four equations, cancel this down and you get the required equation.

3. I think the theorem which states: "ab = pq" must be Ptolemy. But I am not sure about its name.

4. No, this is not Ptolemy's theorem.
For this one, put two intersecting chords in a circle. Call them say AB and PQ and call the point of intersection C.
By the angles on the same arc being equal theorem you have a choice of two sets of similar triangles, ACQ and PCB, or ACP and BCQ.
Choose one of the two, equate corresponding ratios, cross multiply and you have your result.