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2 tangential quadrilaterals from vertices of a quadrilateral & a point on 1 side

Source: Sharygin Geometry Olympiad 2015 Final 8.7

August 1, 2018
geometrycyclic quadrilateraltangential quadrilateral

Problem Statement

Point MM on side ABAB of quadrilateral ABCDABCD is such that quadrilaterals AMCDAMCD and BMDCBMDC are circumscribed around circles centered at O1O_1 and O2O_2 respectively. Line O1O2O_1O_2 cuts an isosceles triangle with vertex M from angle CMDCMD. Prove that ABCDABCD is a cyclic quadrilateral.
(M. Kungozhin)
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