Cousins of Pythagoras: Generalized Quadratic Relations in Triangles with Fixed and Angular Coefficients

This study explores extensions of the classical Pythagorean Theorem by introducing what are termed the “Cousins of Pythagoras”—a set of quadratic side-length relations applicable to both special and general triangles. While the classical relation a^2+b^2=c^2 holds uniquely for right triangles with a fixed coefficient, we derive analogous formulas for equilateral, 30-60-90, and arbitrary triangles. In particular, we present a generalized form involving an angular-dependent coefficient derived from the Law of Cosines:<br>a^2+b^2=K(α)⋅c^2,<br>where K(α)=1+2ab/c^2cos⁡(α).<br>The paper provides step-by-step derivations and examines special cases where this coefficient becomes fixed (e.g., equilateral or right triangles). These results offer a deeper geometric understanding of triangle structures and demonstrate how classical theorems can be broadened into unified frameworks across all triangle types.<br>The author acknowledges the use of OpenAI’s language model for assistance in drafting certain sections of this workl