Lawrence (abangaku) wrote in moppers,

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renegade olympiad problem?

okay, so... anyone know a simple proof of the following fact (which i really find fascinating)?

take a cyclic quadrilateral. there are two ways to divide it into two triangles, using a diagonal. prove that the sum of the two inradii of the triangles is the same, no matter what your choice of diagonal is. (you can easily generalize this to any cyclic polygon divided into triangles.)

it's such a simple fact, i can't believe i haven't seen it in any exposition of theorems. but i've only seen complex proofs of it. i hope it generalizes into some big theory, like inversion. that would be exciting.

i also wonder if the converse is true... if the sums of the two inradii are equal, is the quadrilateral cyclic necessarily?
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I have a reasonably nice proof that only involves angle-chasing and stuff.

The key lemma is that the four incenters form a rectangle with sides parallel to the lines passing through the midpoints of the arcs of the circumcircle cut off by opposite sides (those points are very important in the proof, because they're how you get a hold on the incenter). I never would have thought of this lemma on my own, but I had to prove part of it for something else.
Oh, and Mike claims this is a standard fact taught at MOP, but can't recall the proof. I think I must have slept through that part of the lecture.
Continuing the comment chain, Mike and I want to know if by complex you meant "not simple" or "using complex numbers".