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Existence of tangent circle given <BAC=60

Source: 2024 Israel TST Test 1 P1

August 29, 2023
geometrytangencyTSTcircumcircle

Problem Statement

Triangle ABCABC with ∠BAC=60∘\angle BAC=60^\circ is given. The circumcircle of ABCABC is Ω\Omega, and the orthocenter of ABCABC is HH. Let SS denote the midpoint of the arc BCBC of Ω\Omega which doesn't contain AA. Point PP was chosen on Ω\Omega so that ∠HPS=90∘\angle HPS=90^\circ. Prove that there exists a circle that goes through PP and SS and is tangent to lines ABAB, ACAC.
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