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Geometry Mathley 12.4 orthogonal circles pairwise

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June 7, 2020
circlesConcyclic

Problem Statement

QuadrilateralABCD ABCD has two diagonals AC,BDAC,BD that are mutually perpendicular. Let MM be the Miquel point of the complete quadrilateral formed by lines AB,BC,CD,DAAB,BC,CD,DA. Suppose that LL is the intersection of two circles (MAC)(MAC) and (MBD)(MBD). Prove that the circumcenters of triangles LAB,LBC,LCD,LDALAB,LBC,LCD,LDA are on the same circle called ω\omega and that three circles (MAC),(MBD),ω(MAC), (MBD), \omega are pairwise orthogonal.
Nguyễn Văn Linh
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