Original scientific paper

https://doi.org/10.31896/k.28.2

Miquel's Theorem and its Elementary Geometric Relatives

Gunter Weiss ; University of Technology Vienna, Vienna, Austria *
Boris Odehnal ; University of Applied Arts Vienna, Vienna, Austria

* Corresponding author.

Abstract

The elementary geometric Miquel theorem concerns a triangle ABC and points R, S, T on its sides, and it states that the circles k(ART), k(BRS), k(CST) have a common M, the Miquel point to these givens. Choosing R, S, T in special ways one receives the so-called beermat theorem, the Brocard theorems, and the Steiner Simson-Wallace theorem as special cases of Miquel's theorem. Hereby facts connected with Brocard's theorem follow from properties of Miquel's theorem. If e.g. R, S, T fullfil the Ceva condition, Miquel's construction induces a mapping of the Ceva point to the Miquel point. We discuss this and other mappings, which are natural consequences of Miquel's theorem. Furthermore, if the points R, S, T run through the sides of the triangle such that e.g. the affine ratios ar(ARB), ar(BSC), ar(CTA) are equal, then the corresponding Miquel points M run through the circumcircle of the triangle formed by the Brocard points and the circumcenter of the triangle. Besides these three remarkable points of the triangle, this circle contains several other triangle centers. Even though most of the presented topics are well-known, their mutual connections seem to be not yet considered in standard references on triangle geometry and therefore might justify an additional treatment.

Keywords

Miquel's theorem; Brocard's theorems; theorems of Steiner and Simson-Wallace

Hrčak ID:

323761

URI

https://hrcak.srce.hr/323761

Publication date:

20.12.2024.

Article data in other languages: croatian

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