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Circles Tangent to Sides and Circumcircle

Source: Turkey EGMO TST 2014 P5

August 6, 2016
geometrycircumcircle

Problem Statement

Let ABCABC be a triangle with circumcircle ω\omega and let ωA\omega_A be a circle drawn outside ABCABC and tangent to side BCBC at A1A_1 and tangent to ω\omega at A2A_2. Let the circles ωB\omega_B and ωC\omega_C and the points B1,B2,C1,C2B_1, B_2, C_1, C_2 are defined similarly. Prove that if the lines AA1,BB1,CC1AA_1, BB_1, CC_1 are concurrent, then the lines AA2,BB2,CC2AA_2, BB_2, CC_2 are also concurrent.
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