Charles Thas,  On Some Remarkable Concurrences,

Forum Geometricorum, 2 (2002) 147--149.

Abstract:  In [2], Bruce Shawyer proved the following result. At the midpoint of  each side of a triangle, we construct the line such that the product of the slope of this line and the slope of the side of the triangle is a fixed constant. We show that the three lines obtained are always concurrent. Further, the locus of the points of concurrency is a rectangular hyperbola. This hyperbola intersects the sides of the triangle at the midpoints of the sides, and each side at another point. These three other points, when considered with the vertices of the triangle opposite to the point, form a Ceva configuration. Remarkably, the point of concurrency of these Cevians lies on the circumcircle of the original triangle. Here, we extend these results in the projective plane and give a short synthethic proof.

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