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Common internal tangent passes through a midpoint of a diagonal

Source: Sharygin 2022 Finals 10-11.4

August 2, 2022
geometry

Problem Statement

Let ABCDABCD be a convex quadrilateral with ∠B=∠D\angle B= \angle D. Prove that the midpoint of BDBD lies on the common internal tangent to the incircles of triangles ABCABC and ACDACD.
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