On the Sequence of Pedal Triangles

Although geometers have studied the properties of triangles for over two thousand years, there still remain problems of interest involving operations performed infinitely often. A given triangle T_0 generates a sequence of triangles T_n where T_(n+1) is the pedal triangle of T_n. This sequence was discussed by Hobson (1897, 1925) but, while his formulae for the transition from T_n to T_(n+1) are correct, those for T_n in terms of T_0 are not. Lacking correct formulae, we experimented numerically, taking the angles of T_0 to be integers in degrees. To our surprise the angles in the pedal sequence became periodic with periods of 12 steps. The explanation of this curious fact led to a general investigation of pedal sequences, revealing that (a) the sequence may stop by degeneration of the triangle to a straight segment, (b) the angles may develop any periodicity, or (c) the sequence may proceed to infinity without any periodicity. We give necessary and sufficient conditions on the angles of T_0 corresponding to these options, and discuss the periodic case in some detail