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Complementary or equal angles

Source: 2023 Israel TST Test 2 P3

March 23, 2023
TSTgeometry

Problem Statement

In triangle ABCABC the orthocenter is HH and the foot of the altitude from AA is DD. Point PP satisfies AP=HPAP=HP, and the line PAPA is tangent to (ABC)(ABC). Line PDPD intersects lines AB,ACAB, AC at points X,YX,Y respectively.
Prove that ∠YHX=∠BAC\angle YHX = \angle BAC or ∠YHX+∠BAC=180∘\angle YHX+\angle BAC= 180^\circ.
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