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In the geometry of the triangle, the Feuerbach quadrangle comprises the four points of contact between the nine point circle and the four tritangent circles. By omitting each of the four points in turn, we create four Feuerbach triangles. When the basic triangle is isosceles we find that two of the four points of contact coincide and the resultant single Feuerbach triangle is also isosceles. In the spirit of Steiner-Lehmus we address the converse question – if one of the Feuerbach triangles is isosceles, is the basic triangle necessarily isosceles? The answer is negative.

The celebrated Steiner-Lehmus theorem states that if two internal bisectors of a triangle are equal then the triangle is isosceles. It beautifully illustrates the point that the converse of a theorem may be far more difficult to prove than the direct statement. The literature of the Steiner-Lehmus theorem is considerable with many contributions in the Mathematical Gazette. An interesting variation on this theme may be found by examining the triangle formed by the second intersections of the medians with the circumcircle. Let the median point of a triangle ABC be G and let the medians meet the opposite sides at DEF and the circumcircle again at LMN.