A circle touches the two legs of an angle. How can one draw a line that intersects both legs, such that the circle lies within the triangle with as sides the two legs and the drawn line, and such that the perimeter of the triangle is minimal? Hint: use the sine rule for triangles and do not forget the identity $\sin u + \sin v = 2 \sin\left(\frac{1}{2}(u+v)\right)\cos\left( \frac{1}{2} (u-v) \right)$.
I think I can see the answer geometrically, but I am specifically asked to model the problem with equations. Could anyone please give me some more ideas on where I should start?