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segments from P to the incircle and circumcircle are equal

Source: Sharygin 2006 IX- XI CR 15

August 23, 2019
geometrycircumcircleequal segmentsincircle

Problem Statement

A circle is circumscribed around triangle ABCABC and a circle is inscribed in it, which touches the sides of the triangle BC,CA,ABBC,CA,AB at points A1,B1,C1A_1,B_1,C_1, respectively. The line B1C1B_1C_1 intersects the line BCBC at the point PP, and MM is the midpoint of the segment PA1PA_1. Prove that the segments of the tangents drawn from the point MM to the inscribed and circumscribed circle are equal.
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