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Geometry Mathley 11.4 6 concurrent circles

Source:

June 7, 2020
geometrycircumcirclecirclesconcurrentconcurrent circles

Problem Statement

Let ABCABC be a triangle and PP be a point in the plane of the triangle. The lines AP,BP,CPAP,BP, CP meets BC,CA,ABBC,CA,AB at A1,B1,C1A_1,B_1,C_1, respectively. Let A2,B2,C2A_2,B_2,C_2 be the Miquel point of the complete quadrilaterals AB1PC1BCAB_1PC_1BC, BC1PA1CABC_1PA_1CA, CA1PB1ABCA_1PB_1AB. Prove that the circumcircles of the triangles APA2APA_2,BPB2BPB_2, CPC2CPC_2, BA2CBA_2C, AB2CAB_2C, AC2BAC_2B have a point of concurrency.
Nguyễn Văn Linh
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