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Tangency of circles with "135 degree" angles

Source: Iran Team selection test 2024 - P12

May 19, 2024
geometry

Problem Statement

For a triangle △ABC\triangle ABC with an obtuse angle ∠A\angle A , let E,FE , F be feet of altitudes from B,CB , C on sides AC,ABAC , AB respectively. The tangents from B,CB , C to circumcircle of triangle △ABC\triangle ABC intersect line EFEF at points K,LK , L respectively and we know that ∠CLB=135\angle CLB=135. Point RR lies on segment BKBK in such a way that KR=KLKR=KL and let SS be a point on line BKBK such that KK is between B,SB , S and ∠BLS=135\angle BLS=135. Prove that the circle with diameter RSRS is tangent to circumcircle of triangle △ABC\triangle ABC.
Proposed by Mehran Talaei
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