Triangles with Equivalent Relations Between the Angles and Between the Sides

Abstract

The simplest examples of equivalent relations between angles and sides for a triangle $\triangle ABC$ with sides $a$, $b$, and $c$ are well known, e.g., $$\leqalignno{ \angle A = \angle B &\Leftrightarrow a = b &(1)\cr}$$ and $$\leqalignno{ \angle A = \angle B + \angle C &\Leftrightarrow a^2 = b^2 + c^2 \ ,&(2)\cr}$$ because the angle-relation is equivalent to $\angle A = {\pi \over 2}$.

K. Schwering [7, 8, 9] and J. Heinrichs [3] (see also Dickson [2]) have studied relations of the form $\angle A = n \angle B$ and $\angle A = n \angle B + \angle C$ by the help of trigonometric functions and roots of unity. W. W. Willson [12] and R. S. Luthar [4] have considered the case $n = 2$, and recently J. E. Carroll and K.~Yanosko [1] have generalized to the case of $n$ rational. E. A. Maxwell [5, 6] has considered triangles with $2 \angle A = \angle B + \angle C$. In this paper we present elementary geometric proofs of the following equivalences: $$\leqalignno{ \angle A = 2 \angle B &\Leftrightarrow a^2 = b^2 + bc &(3)\cr \angle A = 2 \angle B + \angle C &\Leftrightarrow a^2 = b^2 + ac &(4)\cr 2 \angle A = \angle B + \angle C &\Leftrightarrow a^2 = b^2 + c^2 - bc &(5)\cr \angle A = 2 (\angle B + \angle C ) &\Leftrightarrow a^2 = b^2 + c^2 + bc &(6)\cr \angle A = 2 (\angle B - \angle C ) &\Leftrightarrow b a^2 = (b-c)(b+c)^2 &(7)\cr}$$

Furthermore, we present the formulas for the complete set of integral solutions for each of the types of triangles.