When a triangle is isosceles?

Abstract

In 1840 Jacob Steiner on Christian Rudolf's request proved that a triangle with two equal bisectors is isosceles. But what about changing the bisectors to cevians? Cevian is any line segment in a triangle with one endpoint on a vertex of the triangle and other endpoint on the opposite side. Not for any pairs of equal cevians the triangle is isosceles. Theorem. If for a triangle ABC there are equal cevians issuing from A and B, which intersect on the bisector or on the median of the angle C, then AC=BC (so the triangle ABC is isosceles). Proposition. Let ABC be an isosceles triangle. Define circle C to be the circle symmetric relative to AB to the circumscribed circle of the triangle ABC. Then the locus of intersection points of pairs of equal cevians is the union of the base AB, the triangle's axis of symmetry, and the circle C.