Darij Grinberg and Alexei Myakishev, A Generalization of the Kiepert Hyperbola,
Forum Geometricorum, 4 (2004) 253--260.

Abstract:   Consider  an arbitrary point P in the plane of triangle ABC with cevian triangle A_1B_1C_1. Erecting similar isosceles triangles on the segments BA_1, CA_1, CB_1, AB_1, AC_1, BC_1, we get six apices. If the apices of the two isosceles triangles with bases BA_1 and CA_1 are connected by a line, and the two similar lines for B_1 and C_1 are drawn, then these three lines form a new triangle, which is perspective to triangle ABC. For fixed P and varying base angle of the isosceles triangles, the perspector draws a hyperbola. Some properties of this hyperbola are studied in the paper.

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