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(AN)(NC) =(CD)(BN) in a cyclic ABCD with (AD)=(BD)

Source: 2013 Dutch BxMO / EGMO TST p5

September 19, 2018
geometrycyclic quadrilateralProductincirclecircumcircle

Problem Statement

Let ABCDABCD be a cyclic quadrilateral for which ∣AD∣=∣BD∣|AD| =|BD|. Let MM be the intersection of ACAC and BDBD. Let II be the incentre of △BCM\triangle BCM. Let NN be the second intersection pointof ACAC and the circumscribed circle of △BMI\triangle BMI. Prove that ∣AN∣⋅∣NC∣=∣CD∣⋅∣BN∣|AN| \cdot |NC| = |CD | \cdot |BN|.
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