Three Coplanar Bisectors in Unit Sphere Construction

Draw a spherical triangle on the surface of the unit sphere centered at . Let be the point opposite on . Let the sides opposite the corresponding vertices be the arcs , , . Then the bisectors of , and (the supplementary angle of ) lie in the same plane.


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Vectors parallel to the bisectors of , , are , , , but , so the vectors are coplanar [3, p. 83].
[1] Wikipedia. "Spherical Law of Cosines." (May 15, 2017) _cosines.
[2] Wikipedia. "Spherical Trigonometry." (May 15, 2017)
[3] V. V. Prasolov and I. F. Sharygin, Problems in Stereometry (in Russian), Moscow: Nauka, 1989.
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