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Geometry Mathley 8.4 concurrent circles

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June 7, 2020
concurrent circlescirclestangent circlesorthocenter

Problem Statement

Let ABCABC a triangle inscribed in a circle (O)(O) with orthocenter HH. Two lines d1d_1 and d2d_2 are mutually perpendicular at HH. Let d1d_1 meet BC,CA,ABBC,CA,AB at X1,Y1,Z1X_1, Y_1,Z_1 respectively. Let A1B1C1A_1B_1C_1 be a triangle formed by the line through X1X_1 perpendicular to BCBC, the line through Y1Y_1 perpendicular to CA, the line through Z1Z_1 perpendicular perpendicular to ABAB. Triangle A2B2C2A_2B_2C_2 is defined in the same manner. Prove that the circumcircles of triangles A1B1C1A_1B_1C_1 and A2B2C2A_2B_2C_2 touch each other at a point on (O)(O).
Nguyễn Văn Linh
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