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Geometry Mathley 3.3 collinear, 4 lines concurrent

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June 7, 2020
geometryconcurrentcollinearcircumcircle

Problem Statement

A triangle ABCABC is inscribed in circle (O)(O). P1,P2P1, P2 are two points in the plane of the triangle. P1A,P1B,P1CP_1A, P_1B, P_1C meet (O)(O) again at A1,B1,C1A_1,B_1,C_1 . P2A,P2B,P2CP_2A, P_2B, P_2C meet (O)(O) again at A2,B2,C2A_2,B_2,C_2. a) A1A2,B1B2,C1C2A_1A_2, B_1B_2, C_1C_2 intersect BC,CA,ABBC,CA,AB at A3,B3,C3A_3,B_3,C_3. Prove that three points A3,B3,C3A_3,B_3,C_3 are collinear. b) PP is a point on the line P1P2.A1P,B1P,C1PP_1P_2. A_1P,B_1P,C_1P meet (O) again at A4,B4,C4A_4,B_4,C_4. Prove that three lines A2A4,B2B4,C2C4A_2A_4,B_2B_4,C_2C_4 are concurrent.
Trần Quang Hùng
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