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all triangles DPQ by moving point M are similar to each other

Source: IV Soros Olympiad 1997-98 R2 11.6 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

June 1, 2024
geometrysimilar trianglescyclic quadrilateral

Problem Statement

It is known that the bisector of the angle ∠ADC\angle ADC of the inscribed quadrilateral ABCDABCD passes through the center of the circle inscribed in the triangle ABCABC. Let MM be an arbitrary point of the arc ABCABC of the circle circumscribed around ABCDABCD. Denote by PP and QQ the centers of the circles inscribed in the triangles ABMABM and BCMBCM. Prove that all triangles DPQDPQ obtained by moving point MM are similar to each other. Find the angle ∠PDQ\angle PDQ and ratio BP:PQBP : PQ if ∠BAC=α\angle BAC = \alpha, ∠BCA=β\angle BCA = \beta
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