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Prove that lines $AH$, $XY$ and $DE$ are concurrent

Source: Moldova TST 2022

April 1, 2022
geometry

Problem Statement

In the acute triangle ABCABC the point MM is on the side BCBC. The inscribed circle of triangle ABMABM touches the sides BMBM, MAMA and ABAB in points DD, EE and FF, and the inscribed circle of triangle ACMACM touches the sides CMCM, MAMA and ACAC in points XX, YY and ZZ. The lines FDFD and ZXZX intersect in point HH. Prove that lines AHAH, XYXY and DEDE are concurrent.
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