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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