Relationships Between Six Circumcircles

TL;DR

This paper explores the geometric relationships between the radii of circumcircles and the positions of circumcenters of six small triangles formed inside a larger triangle by cevians through an interior point.

Contribution

It provides new theorems linking the circumcircle radii and circumcenters of the six sub-triangles created by cevians through an interior point.

Findings

Derived formulas relating circumcircle radii

Established relationships between circumcenters

Enhanced understanding of internal cevian configurations

Abstract

If $P$ is a point inside $\triangle ABC$, then the cevians through $P$ divide $\triangle ABC$ into six small triangles. We give theorems about the relationships between the radii of the circumcircles of these triangles. We also state some results about the relationships between the circumcenters of these triangles.