Pedal triangles, self-adjoint affine maps and a two-thirds similarity

The affine map sending a triangle ABC to an inscribed triangle A1B1C1 is self-adjoint exactly when the perpendiculars to the sides at A1, B1, C1 concur, i.e. when A1B1C1 is the pedal triangle of a point M. It is then a pure strain with principal factors (−1 ± OM/R)/2 along the Simson-line directions of the points where OM meets the circumcircle. Consequently, the points dividing AA1, BB1, CC1 in the ratio 2:1 form a triangle inversely similar to ABC with ratio OM/(3R), a property that characterizes pedal triangles.