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two circumscribed quadrilateral relates to touchpoints of incircle

Source: Sharygin Geometry Olympiad 2017 First Round P15 grades 9-11

July 22, 2018
geometrycircumscribed quadrilateralperpendicularincircle

Problem Statement

Let ABCABC be an acute-angled triangle with incircle ω\omega and incenter II. Let ω\omega touch AB,BCAB, BC and CACA at points D,E,FD, E, F respectively. The circles ω1\omega_1 and ω2\omega_2 centered at J1J_1 and J2J_2 respectively are inscribed into ADIFDIF and BDIEBDIE. Let J1J2J_1J_2 intersect ABAB at point MM. Prove that CDCD is perpendicular to IMIM.
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