208. An interesting geometrical inequality.

by Virgil Nicula, Jan 14, 2011, 8:14 AM

Proposed problem. Let $ P$ be an arbitary point inside to $ \triangle ABC$ . Denote the second intersection $ A_1$ of the line $ AP$ with

the circumcircle of $ \triangle PBC$ . Define similarly $ B_{1}$ , $ C_{1}$ . Prove that $ \prod\left(1 + 2\cdot\frac {PA}{PA_{1}}\right)\ge 8$ and $ \sum\frac {PA}{PA_1}\ge\frac 32$ .
Proof I.

Proof II.
This post has been edited 31 times. Last edited by Virgil Nicula, Nov 22, 2015, 4:28 PM

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