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parallel lines related to touchpoints of excircle and incircle

Source: 2011 Sharygin Geometry Olympiad Correspondence Round P11

April 4, 2019
geometryparallelincircleexcircle

Problem Statement

The excircle of right-angled triangle ABCABC (∠B=90o\angle B =90^o) touches side BCBC at point A1A_1 and touches line ACAC in point A2A_2. Line A1A2A_1A_2 meets the incircle of ABCABC for the first time at point A′A', point C′C' is defined similarly. Prove that AC∣∣A′C′AC||A'C'.
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