Spherical triangles of area Pi

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April 6, 2018 beni22sof Leave a comment Go to comments

Recently I stumbled upon this page and found out a very nice result:

If a spherical triangle has area $\pi$ then four copies of it can tile the sphere.

Here we are talking about triangles on the unit sphere whose edges are geodesics. The above result is a simple consequence of the following facts:

If a spherical triangle has angles $\alpha,\beta,\gamma$ then its area is $\alpha+\beta+\gamma-\pi$ . Therefore if a triangle has area $\pi$ , then $\alpha+\beta+\gamma = 2\pi$ .
If $ABC$ is a triangle of area $\pi$ and $D$ is obtained by symmetrizing $A$ with respect to the midpoint of $BC$ on the sphere, then $DCB$ and $ABC$ are congruent triangles.
Using angles and the fact that on the sphere similar triangles are congruent, we obtain that the triangles $ABC,BCD,CDA,DAC$ are all congruent.