Is an equilateral triangle always equiangular?

Answer 1

Yes, the angles at the vertices are each 60 degrees (total 180 degrees, as do the sum of the angles in all triangles).

The Law of Sines (see related link) can be used to show this:

a/sin(A) = b/sin(B) = c/sin(C), where a,b,c are the side lengths, and A,B,C are the angles (opposite of sides a,b,c respectively).

Since a = b = c for equilateral, then sin(A) = sin(B) = sin(C). Now you can have two different angles which give the same sine, say for example if A=80°, and B=100°, then sin(80°) = sin(100°), but then because they must add to 180°, then angle C must be 0°, so it's not a triangle anymore, so A and B must be equal, then in the same way it can be shown that C must also equal A and B. Since all three angles are equal, and they must add to 180°, then the angle equals 180° / 3 = 60°.

Note that for any angle x, the other angle which has the same sine is [180° - x].

This means, of course, that every equilateral triangle, is simply a smaller or larger version of every other equilateral triangle (they are similar).