Triangle conics and cubics

TL;DR

This paper develops an algorithm for creating triangle conics and cubics passing through triangle centers, revealing new geometric objects and properties, with implications in Euclidean and projective geometry.

Contribution

It introduces a novel algorithm for constructing triangle conics and cubics through centers, discovering new curves and their properties in Euclidean and projective geometry.

Findings

Derived new triangle conics and cubics.

Established properties of these curves in Euclidean space.

Connected results to projective geometry corollaries.

Abstract

This is a paper about triangle cubics and conics in classical geometry with elements of projective geometry. In recent years, N.J. Wildberger has actively dealt with this topic using an algebraic perspective. Triangle conics were also studied in detail by H.M. Cundy and C.F. Parry recently. The main task of the article was to develop an algorithm for creating curves, which pass through triangle centers. During the research, it was noticed that some different triangle centers in distinct triangles coincide. The simplest example: an incenter in a base triangle is an orthocenter in an excentral triangle. This was the key for creating an algorithm. Indeed, we can match points belonging to one curve (base curve) with other points of another triangle. Therefore, we get a new intersting geometrical object. During the research were derived number of new triangle conics and cubics, were considered their properties in Euclidian space. In addition, was discussed corollaries of the obtained theorems in projective geometry, what proves that all of the descovered results could be transfered to the projeticve plane.