Lemma › Problems › Prove Ptolemy's inequality: for any four points A,B,C,D in the plane,…
Classic olympiad problem · Geometry

Prove Ptolemy's inequality: for any four points A,B,C,D in the plane,…

Prove Ptolemy's inequality: for any four points A,B,C,DA,B,C,D in the plane, AC⋅BD≤AB⋅CD+BC⋅ADAC \cdot BD \le AB\cdot CD + BC \cdot AD, with equality exactly when ABCDABCD is cyclic in that order.
Topic: Geometry Difficulty: 5/6 Expected time: ~45 min Progressive hints on Lemma: 3

This is a proof problem — the answer is an argument, not a number. The solution is deliberately not posted here: reading a solution you didn't fight for teaches almost nothing. On Lemma you attempt it cold, take one of the 3 progressive hints only when genuinely stuck, then compare your proof against a full walkthrough and mark yourself with the same rubric a competition coordinator would use.

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