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Journal for Geometry and Graphics 17 (2013), No. 1, 021--030
Copyright Heldermann Verlag 2013

Equioptic Points of a Triangle

Boris Odehnal
Ordinariat für Geometrie, Universität für Angewandte Kunst, Oskar-Kokoschka-Platz 2, 1010 Wien, Austria

The locus of points where two non-concentric circles c1 and c2 are seen under equal angles is the equioptic circle e. The equioptic circles of the excircles of a triangle Δ have a common radical axis r. Therefore the excircles of a triangle share up to two real points, i.e., the equioptic points of Δ from which the circles can be seen under equal angles. The line r carries a lot of known triangle centers. Further we find that any triplet of circles tangent to the sides of Δ has up to two real equioptic points. The three radical axes of triplets of circles containing the incircle are concurrent in a new triangle center.

Keywords: Triangle, excircle, incircle, equioptic circle, equioptic points, center of similarity, radical axis.

MSC: 51M04

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