Menelaus' and Ceva's Theorem for Spherical Triangle

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Draw a spherical triangle on the surface of a unit sphere centered at . Let the sides opposite the corresponding vertices be the arcs , , and contain the points , , . Menelaus's theorem for a spherical triangle states:

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The rays , , are on the same plane if and only if

.

Ceva's theorem for a spherical triangle states:

The planes determined by pairs of rays , and go through the same ray () if and only if

.

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


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Details

A proof of these theorems can be found in [3, pp. 85–86].

References

[1] Wikipedia. "Spherical Law of Cosines." (Mar 20, 2017) en.wikipedia.org/wiki/Spherical_law_of _cosines.

[2] Wikipedia. "Spherical Trigonometry." (Mar 20, 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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