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

Text
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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 |
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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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