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Geometry Mathley 5.4 5 circles concurrent

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June 7, 2020
concurrentconcurrent circlescirclesgeometry

Problem Statement

Let ABCABC be a triangle inscribed in a circle (O)(O). Let PP be an arbitrary point in the plane of triangle ABCABC. Points A′,B′,C′A',B',C' are the reflections of PP about the lines BC,CA,ABBC,CA,AB respectively. XX is the intersection, distinct from AA, of the circle with diameter APAP and the circumcircle of triangle AB′C′AB'C'. Points Y,ZY,Z are defined in the same way. Prove that five circles (O),(AB′C′)(O), (AB'C'), (BC′A′),(CA′B′),(XYZ)(BC'A'), (CA'B'), (XY Z) have a point in common.
Nguyễn Văn Linh
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