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All 6 vertices of the two triangles lie on single circle

Source: IMO Shortlist 1995, G4, Iran PPCE 1997, P2

August 10, 2008
trigonometrygeometrycircumcirclehexagonIMO Shortlist

Problem Statement

An acute triangle ABC ABC is given. Points A1 A_1 and A2 A_2 are taken on the side BC BC (with A2 A_2 between A1 A_1 and C C), B1 B_1 and B2 B_2 on the side AC AC (with B2 B_2 between B1 B_1 and A A), and C1 C_1 and C2 C_2 on the side AB AB (with C2 C_2 between C1 C_1 and B B) so that
\angle AA_1A_2 \equal{} \angle AA_2A_1 \equal{} \angle BB_1B_2 \equal{} \angle BB_2B_1 \equal{} \angle CC_1C_2 \equal{} \angle CC_2C_1.
The lines AA1,BB1, AA_1,BB_1, and CC1 CC_1 bound a triangle, and the lines AA2,BB2, AA_2,BB_2, and CC2 CC_2 bound a second triangle. Prove that all six vertices of these two triangles lie on a single circle.
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