Triangles in the Poincare Disc

This demo shows the existance of a triangle in the Poincare Disc for any three angles t1, t2, t3 with sum < pi. In the "Set the thetas" window, fix t1, t2 and the sum. Then move the red hotspot in the Poincare Disc window (varying u from 0 to 1) to find two values of f(u) that give triangles with third angle greater and less than t3. Since f is a continuous function of u, there exists a triangle with third angle t3 by the Intermediate Value Theorem.