Copyright © University of Cambridge. All rights reserved.
You only need elementary
trigonometry and scalar products
Given any rightangled triangle $\Delta ABC$ on a sphere of
unit radius, right angled at $A$, and with lengths of sides $a, b$
and $c$, then Pythagoras' Theorem in Spherical Geometry is $$\cos a
= \cos b \cos c.$$ Prove this result.
Find a triangle containing three right angles on the surface
of a sphere of unit radius. What are the lengths of the sides of
your triangle?
Use the Pythagoras' Theorem result above to prove that all
spherical triangles with three right angles on the unit sphere are
congruent to the one you found.
