A Generalization of Certain Remarkable Points of the Triangle Geometry

In this article we prove a theorem that will generalize the concurrence theorems that are leading to the Franke’s point, Kariya’s point, and to other remarkable points from the triangle geometry.

💡 Research Summary

The paper presents a unifying concurrency theorem that subsumes several well‑known remarkable points of triangle geometry, notably Franke’s point, Kariya’s point, as well as other classical centers such as the Euler and Nagel points. After a concise review of the historical development of concurrency results—primarily Ceva’s and Menelaus’ theorems—the author introduces a generalized configuration: for a given triangle (ABC) three arbitrary points (P, Q, R) are placed on the sides (BC, CA, AB) (or their extensions). Through each of these points two circles are drawn: one passing through the corresponding side’s endpoints (an “excircle” of the side) and another touching the side at the chosen point (an “incircle” relative to that side).

Using barycentric coordinates ((\alpha:\beta:\gamma)) for (P, Q, R), the equations of all six circles are expressed in a common linear form. The core of the work is Theorem 1, which states that if the points satisfy a generalized Ceva‑type product
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