Triangle Centers Relative to Their Cevian, Circumcevian, and Pedal Triangles

Abstract

For a triangle center, the paper asks whether the same Euclidean point is classified as another triangle center relative to its cevian, circumcevian, or pedal triangle. It gives a self-contained circumcevian-pedal transfer theorem, an oriented parameter-reversal theorem for the family F_λ, a general near-equilateral obstruction to positive-Heron fixed-point claims, and an exact proof of the high-index fixed relation M₂: X₆₇₃₇₁ → X₆₇₃₇₁.

A computational census of a local Encyclopedia of Triangle Centers snapshot through X₇₂₈₀₇ reports the tested scope, exact survivors, branch-sensitive failures, and unresolved candidates. The paper also records the classical status of the transfer, Steiner-focus, and X₁₈₇ relations, with reproducible algebraic checks.