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danube 3 common chords' concurrency

Source: Danube 2008 P2

September 8, 2018
geometryChordsmidpointcirclesconcurrencyconcurrent

Problem Statement

In a triangle ABCABC let A1A_1 be the midpoint of side BCBC. Draw circles with centers A,A1A, A1 and radii AA1,BCAA_1, BC respectively and let A′A′′A'A'' be their common chord. Similarly denote the segments B′B′′B'B'' and C′C′′C'C''. Show that lines A′A′′,B′B′′′A'A'', B'B''' and C′C′′C'C'' are concurrent.
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