How can you prove that a vertex of a triangle on the poincar

How can you prove that a vertex of a triangle on the poincare\' disk has a measure of 0 degrees?

Solution

Consider the points A = (1,0), B = (cos120,sin120), and C = (cos240,sin240) on the unit circle. If we construct Poincar´e circles through each pair of points, we discover that the angle of each vertex of the triangle ABC has measure zero. Of course, these points lie on the circle, and are therefore not technically inside the Poincar´e open disk—such points on the unit circle are called ideal points. None-the-less, if we take points near them, we see that we can make a triangle whose angle sum is as close to zero .