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
Snapshots
Details
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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