Original scientific paper

Incenter Symmetry, Euler Lines, and Schiffler Points

Nguyen Le ; San Francisco State University, San Francisco, USA
Norman John Wildberger orcid.org/0000-0003-3503-6495 ; School of Mathematics and Statistics UNSW, Sydney, Australia

Abstract

We look at the four-fold symmetry given by the Incenter quadrangle of a triangle, and the relation with the cirumcircle, which in this case is the nine-point conic of the quadrangle. By investigating Euler lines of Incenter triangles, we show that the classical Schiffler point extends to a set of four Schiffler points, all of which lie on the Euler line. We discover also an additional quadrangle of Incenter Euler points on the circumcircle and investigate its interesting diagonal triangle. The results are framed in purely algebraic terms, so hold over a general bilinear form. We present also a mysterious case of apparent symmetry
breaking in the Incenter quadrangle.

Keywords

triangle geometry; Euclidean geometry; rational trigonometry; bilinear form; Schiffler points; Euler lines; Incenter hierarchy; circumcircles

Hrčak ID:

174092

URI

https://hrcak.srce.hr/174092

Publication date:

16.1.2017.

Article data in other languages: croatian

Visits: 2.406 *