Let ABC be a triangle with H and O being its orthocenter and circumcenter, respectively. If the three triangles A′BC, AB′C, and ABC′, constructed on the sides of triangle ABC as bases, are similar, isosceles, and similarly situated, then the lines AA′, BB′, and CC′ are concurrent. Let P be the point of concurrency and let Q be the isogonal conjugate of P. Let the line PQ meet the Euler line of triangle ABC at K. Then KH/HO = f(α), where α is the base angle of the isosceles triangles. This means that if α is fixed, then the ratio KH/HO is a constant independent of the reference triangle ABC.
