Problem

The key is to expand the left side and recognize that it contains dot products of angle sums. When you square both sums and add them, you'll get terms involving $\cos(A-B)$, $\cos(B-C)$, and $\cos(C-A)$—which are maximized when the angles are equal. Since the expression equals 9 (its theoretical maximum), this forces a very specific constraint on the triangle's angles that only finitely many configurations can satisfy.

The key is to expand the left side and recognize that it contains dot products of angle sums. When you square both sums and add them, you'll get terms involving $\cos(A-B)$, $\cos(B-C)$, and $\cos(C-A)$—which are maximized when the angles are equal. Since the expression equals 9 (its theoretical maximum), this forces a very specific constraint on the triangle's angles that only finitely many configurations can satisfy.