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AN x NC = CD x BN wanted, inside cyclic ABCD with AD=BD

Source: 2012 Croatia MO p3

August 5, 2020
geometrycyclic quadrilateralequal segments

Problem Statement

Let ABCDABCD be a cyclic quadrilateral such that ∣AD∣=∣BD∣|AD| =|BD| and let MM be the intersection of its diagonals. Furthermore, let NN be the second intersection of the diagonal ACAC with the circle passing through points B,MB, M and the center of the circle inscribed in triangle BCMBCM. Prove that AN⋅NC=CD⋅BNAN \cdot NC = CD \cdot BN
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