Τρίτη, 17 Οκτωβρίου 2017

2003 IMO Shortlist G3

Let $ABC$ be a triangle and let $P$ be a point in its interior. Denote by $D, E, F$ the feet of the perpendiculars from $P$ to the lines $BC, CA, AB$, respectively. Suppose that
$AP^2 + PD^2 = BP^2 + PE^2 = CP^2 + PF^2$.

Denote by $I_A, I_B , I_C$ the excentres of the triangle $ABC$. Prove that $P$ is the circumcentre of the triangle $I_AI_BI_C$ .



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