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Orthocenter of triangle formed by polars is Nagel point

Source: KoMaL A. 828

June 11, 2022
geometry

Problem Statement

Triangle ABCABC has incenter II and excircles ΩA\Omega_A, ΩB\Omega_B, and ΩC\Omega_C. Let ℓA\ell_A be the line through the feet of the tangents from II to ΩA\Omega_A, and define lines ℓB\ell_B and ℓC\ell_C similarly. Prove that the orthocenter of the triangle formed by lines ℓA\ell_A, ℓB\ell_B, and ℓC\ell_C coincides with the Nagel point of triangle ABCABC.
(The Nagel point of triangle ABCABC is the intersection of segments ATAAT_A, BTBBT_B, and CTCCT_C, where TAT_A is the tangency point of ΩA\Omega_A with side BCBC, and points TBT_B and TCT_C are defined similarly.)
Proposed by Nikolai Beluhov, Bulgaria
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