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Circles through A touching sides of the triangle

Source: Czech-Polish-Slovak Match 2015, Problem 5

June 19, 2015
geometry

Problem Statement

Let ABCABC be an acute triangle, which is not equilateral. Denote by OO and HH its circumcenter and orthocenter, respectively. The circle kk passes through BB and touches the line ACAC at AA. The circle ll with center on the ray BHBH touhes the line ABAB at AA. The circles kk and ll meet in XX (X≠AX\ne A). Show that ∠HXO=180∘−∠BAC\angle HXO=180^\circ-\angle BAC.
Proposed by Josef Tkadlec
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