TITLE: Triangle with Ratio of Sides Equal to Ratio of Angles
QUESTION [5 upvotes]: In an equilateral triangle, the side lengths are in ratio 1:1:1, as are the angle measures.
Are there also non-equilateral triangles in which the ratio of the side lengths is the same as the ratio of the angle measures? If so, what is an example of such a triangle?

REPLY [3 votes]: The equilateral triangle is the only such case.
Let $A$ and $B$ be two angles, and $a$ and $b$ be the respective opposite sides. Then 
$$
\frac{\sin A}{\sin B} = \frac a b
$$
so if your ratios match, then
$$
\frac{\sin A}{\sin B} = \frac a b
$$
$$
\frac{\sin A}{A} = \frac{\sin B}{B} 
$$
So to have a triangle where $A \neq B$ you require two distinct positive points, both  on the curve $f(x) = \frac{\sin x}x$ , having the same value of $f(x)$.  For this to happen one of the points must be greater than $\pi$ and that cannot be an angle in a triangle.
So no two angles of such a triangle can be unequal, thus it must be equilateral.