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Problem 2345

IMO Shortlist mid-range; USAMO P2/P5

Geometry Difficulty 8.0 Prove it KöMaL problem A · Hungary · 2022

Triangle $ABC$ has incenter $I$ and excircles $\Omega_A$ , $\Omega_B$ , and $\Omega_C$ . Let $\ell_A$ be the line through the feet of the tangents from $I$ to $\Omega_A$ , and define lines $\ell_B$ and $\ell_C$ similarly. Prove that the orthocenter of the triangle formed by lines $\ell_A$ , $\ell_B$ , and $\ell_C$ coincides with the Nagel point of triangle $ABC$ .

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Source: KöMaL, licensed Rights held by KöMaL and the MATFUND Foundation. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project. Solutions are the publisher's, linked not copied.