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(CSH) touches the excirlce in right triangle

Source: Oral Moscow Geometry Olympiad 2024, 10-11.3

September 3, 2024
geometryangle bisectorcircumcircle

Problem Statement

The hypotenuse ABAB of a right-angled triangle ABCABC touches the corresponding excircle ω\omega at point TT. Point SS is symmetrical TT relative to the bisector of angle CC, CHCH is the height of the triangle. Prove that the circumcircle of triangle CSHCSH touches the circle ω\omega.
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