Demonstration:Sum of the angles of the hyperbolic triangle
- Pellumb Kllogjeri
In figure is shown a hyperbolic triangle with sides DG, DF and GF that are arches with centre C, E and H respectively. The tangents are not shown for reasons of clarity of the figure but, someone wants them let their display be done. In figure are shown the exterior angles (denoted by α,β and γ) of the angles between two tangents passing through the vertices. The sum of the interior angles of the hyperbolic triangle can be computed using the expression: δ=540°-(α+ β+ γ). To bring the triangle to sight use the GeoGebra tools.
How are determined the centers of the arches? Try to move away the centers, one at a time, from the arches and observe the values of δ. What is your observation: how does the value of δ relate to the movements of the centers? Can you explain why the sum of the interior angles is less than 180°? When this sum approaches the value 180°?