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(EXF) tangent to gamma

Source: Own. IMO 2022 Malaysian Training Camp 1

February 27, 2022
geometry

Problem Statement

Let ABCDABCD be a circumscribed quadrilateral with incircle γ\gamma. Let AB∩CD=E,AD∩BC=F,AC∩EF=K,BD∩EF=LAB\cap CD=E, AD\cap BC=F, AC\cap EF=K, BD\cap EF=L. Let a circle with diameter KLKL intersect γ\gamma at one of the points XX. Prove that (EXF)(EXF) is tangent to γ\gamma.
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