On the Sequence of Pedal Triangles

Kingston, J. G. and Synge, J. L. (1983) On the Sequence of Pedal Triangles. (Preprint)

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Abstract

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.

Item Type: Article
Divisions: School of Theoretical Physics > Preprints
Date Deposited: 06 Jul 2018 13:01
Last Modified: 24 Jul 2018 14:04
URI: http://dair.dias.ie/id/eprint/866

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