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Shouting inversion but has an elegant solution with La Hire

Source: 2023 Turkey NMO 2nd Round P2

December 21, 2023
geometrycircumcirle

Problem Statement

Let ABCABC be a triangle and PP be an interior point. Let ωA\omega_A be the circle that is tangent to the circumcircle of BPCBPC at PP internally and tangent to the circumcircle of ABCABC at A1A_1 internally and let ΓA\Gamma_A be the circle that is tangent to the circumcircle of BPCBPC at PP externally and tangent to the circumcircle of ABCABC at A2A_2 internally. Define B1B_1, B2B_2, C1C_1, C2C_2 analogously. Let OO be the circumcentre of ABCABC. Prove that the lines A1A2A_1A_2, B1B2B_1B_2, C1C2C_1C_2 and OPOP are concurrent.
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