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criterion for tangential ABCD to be cyclic related to midpoints of diagonals

Source: 2011 Sharygin Geometry Olympiad Correspondence Round P20

April 6, 2019
geometrycyclic quadrilateraltangential quadrilateralmidpointsdiagonals

Problem Statement

Quadrilateral ABCDABCD is circumscribed around a circle with center II. Points MM and NN are the midpoints of diagonals ACAC and BDBD. Prove that ABCDABCD is cyclic quadrilateral if and only if IM:AC=IN:BDIM : AC = IN : BD.
Nikolai Beluhov and Aleksey Zaslavsky
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