Professional paper

Two Steiner theorems about complete quadrilaterals

Vladimir Volenec ; Matematički odsjek PMF-a, Sveučilište u Zagrebu

Abstract

In this paper, we study a complete quadrilateral in the Euclidean plane. The quadrilateral has a lot of interesting properties. Here we prove two claims, which were published by the great geometer Jakob
Steiner in 1827, without proof. In a complete quadrilateral, the bisectors of angles are concurrent at 16 points, which are the incenters and excenters of the four triangles. Steiner asserted that these 16 intersections lie four by four on a total of eight circles, each of them on two of these circles. Steiner also found out that these eight circles form two quadruplets, so that each circle from one quadruplet is orthogonal to each circle from the other quadruplet. The claims have been proven many times since then, and here we give one of their proofs. The claims have since been proven many times, and here we will give one of their proofs.

Keywords

complete quadrilateral, cyclic quadrilateral, circle

Hrčak ID:

254601

URI

https://hrcak.srce.hr/254601

Publication date:

15.1.2021.

Article data in other languages: croatian

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