prove that the sum of the angles in any triangle is less than 180 in hyperbolic geometry (or poincare model).

Question

We could use poincare disc model as a hyperbolic geometry model.

I have difficulty understanding poincare disc model. So is there someone to help?

Answer 1

You can choose any point within a triangle, and move it to the center of the disk model using a congruence which preserves all angles. Then you have a triangle whose edges are arcs of circles that are orthogonal to the unit circle. As such, they curve inward from the Euclidean connection of the corners. Therefore the hyperbolic angles are smaller than the Euclidean ones.