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cyclic PQRS related to midpoints and touchpoints of incircle of ABCD

Source: Sharygin 2011 Final 10.2

March 31, 2019
geometrycyclic quadrilateraltangential quadrilateralmidpoints

Problem Statement

Quadrilateral ABCDABCD is circumscribed. Its incircle touches sides AB,BC,CD,DAAB, BC, CD, DA in points K,L,M,NK, L, M, N respectively. Points A′,B′,C′,D′A', B', C', D' are the midpoints of segments LM,MN,NK,KLLM, MN, NK, KL. Prove that the quadrilateral formed by lines AA′,BB′,CC′,DD′AA', BB', CC', DD' is cyclic.
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