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prove AIP and omega cut off equal chords on AD

Source: Sharygin Finals 2023 9.4

August 2, 2023
geometrySharygin Geometry OlympiadSharygin 2023

Problem Statement

The incircle ω\omega of a triangle ABCABC centered at II touches BCBC at point DD. Let PP be the projection of the orthocenter of ABCABC to the median from AA. Prove that the circle AIPAIP and ω\omega cut off equal chords on ADAD.
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