On Ceva points of (almost) equilateral triangles (Q1998888)

The paper under review proves the infinitude of Ceva points on equilateral and almost equilateral triangles that are also rational. The pleasing central idea of the proof is to construct a parameter space that turns out to be an elliptic surface of positive rank. A Ceva point of a triangle is a point whose three cevians have rational length. The cevians of a point \(P\) are the three line segments that join a vertex of the triangle to the opposite side, and whose corresponding lines contain \(P\). An almost equilateral triangle is a triangle whose side lengths are three consecutive integers. A triangle is rational if and only if its side lengths are rational. In fact, the authors prove more than this, namely, that all but finitely many cevians of rational length contain infinitely many Ceva points.