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Three Coplanar Bisectors in Unit Sphere Construction

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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.

Contributed by: Izidor Hafner (May 2017)
Open content licensed under CC BY-NC-SA

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Vectors parallel to the bisectors of , , are , , , but , so the vectors are coplanar [3, p. 83].

References

[1] Wikipedia. "Spherical Law of Cosines." (May 15, 2017) en.wikipedia.org/wiki/Spherical_law_of _cosines.

[2] Wikipedia. "Spherical Trigonometry." (May 15, 2017) en.wikipedia.org/wiki/Spherical_trigonometry.

[3] V. V. Prasolov and I. F. Sharygin, Problems in Stereometry (in Russian), Moscow: Nauka, 1989.

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