Nikolaos Dergiades and Paul Yiu,
Antiparallels and concurrent Euler lines,
Forum Geometricorum, 4 (2004) 1--20.
Abstract: We study the condition for concurrency of the Euler lines of
the three triangles each bounded by two sides of a reference triangle
and an antiparallel to the third side. For example, if the
antiparallels are concurrent at P and the three Euler lines are
concurrent at Q, then the loci of P and Q are respectively the tangent
to the Jerabek hyperbola at the Lemoine point, and the line parallel to
the Brocard axis through the inverse of the deLongchamps point in the
circumcircle. We also obtain an interesting cubic as the locus of
the point P for which the three Euler lines are concurrent when the
antiparallels are constructed through the vertices of the cevian
triangle of P.
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