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Circles Through Fixed Point

Source: Germany TST 2013 P3

April 14, 2020
geometrycircumcirclegeometric transformationreflection

Problem Statement

Let ABCABC be an acute-angled triangle with circumcircle ω\omega. Prove that there exists a point JJ such that for any point XX inside ABCABC if AX,BX,CXAX,BX,CX intersect ω\omega in A1,B1,C1A_1,B_1,C_1 and A2,B2,C2A_2,B_2,C_2 be reflections of A1,B1,C1A_1,B_1,C_1 in midpoints of BC,AC,ABBC,AC,AB respectively then A2,B2,C2,JA_2,B_2,C_2,J lie on a circle.
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