Show that in any circumscribed polygon, there exist three sides which could form a triangle.
Been on it for a while starting with quadrilaterals and trying to connect some properties and proved for quadrilaterals. How to prove the general case.
2026-03-26 07:34:54.1774510494
Beautiful property of every single circunscribed polygon ever
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Your claim follows immediately from taking the longest side and the two sides next to it, by using the basic property that the two tangent line segments from a point to a circle are of the same length.
[My original answer below was due to my misunderstanding of "circumscribed".]
I have proven that your proof for quadrilaterals is wrong, and hence your property is too beautiful to be true. Let $ABCD$ be a trapezium such that $AD \parallel CD$ and $(\overline{AB},\overline{CD},\overline{BC},\overline{AD}) = (9,6,2,2)$. Clearly $ABCD$ exists and is cyclic, but no three of its sides can be arranged to form a triangle.