Concurrency of Four Euler Lines,

Forum Geometricorum, 1 (2001) 59 -- 68.

Abstract: Using tripolar coordinates, we prove that if P is a point in the plane of triangle ABC such that the Euler lines of triangles PBC, APC and ABP are concurrent, then their intersection lies on the Euler line of triangle ABC. The same is true for the Brocard axes and the lines joining the circumcenters to the respective incenters. We also prove that the locus of P for which the four Euler lines concur is the same as that for which the four Brocard axes concur. These results are extended to a family L_n of lines through the circumcenter. The locus of P for which the four L_n lines of ABC, PBC, APC and ABP concur is always a curve through 15 finite real points, which we identify.

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