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Rhombus and insimilicenters

Source: ISL 2020 G5

July 20, 2021
geometryrhombusgeometric transformationIMO ShortlistIMO Shortlist 2020

Problem Statement

Let ABCDABCD be a cyclic quadrilateral. Points K,L,M,NK, L, M, N are chosen on AB,BC,CD,DAAB, BC, CD, DA such that KLMNKLMN is a rhombus with KL∥ACKL \parallel AC and LM∥BDLM \parallel BD. Let ωA,ωB,ωC,ωD\omega_A, \omega_B, \omega_C, \omega_D be the incircles of △ANK,△BKL,△CLM,△DMN\triangle ANK, \triangle BKL, \triangle CLM, \triangle DMN.
Prove that the common internal tangents to ωA\omega_A, and ωC\omega_C and the common internal tangents to ωB\omega_B and ωD\omega_D are concurrent.
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