My homework question is what is the product of rotations through opposite angles $\alpha, -\alpha$ about two distinct points. The answer is clearly a translation, but I'm not sure how to prove it. My idea on how to prove this is to draw a triangle, ABC, and its two rotations, A'B'C' and A"B"C", and then show AA", BB", CC" are parallel, however I don't know how to get to this point.
A rotation can be represented as as reflection in two lines intersecting in the point of reflection. These lines can be chosen arbitrarily, except that he angle between them must be half the required angle of rotation. So you need four lines. Choose two of these coincident. Reflection in coincident lines cancel. The remaining two lines are parallel. Reflection in parallel lines is a translation.
Hint: Distance and orientation preserving transformations of the plane are a composition of a translation with a rotation with a translation, or in other words, rotation about some fixed point composed with a translation. Furthermore, when you take the product of two such distance and orientation preserving transformations, the rotation angle you get is the sum of the two rotation angles. So the angle of rotation you get is $0$, so what can the transformation be?