11284

Four Theorems on Spherical Triangles

Draw a spherical triangle on the surface of a unit sphere centered at . Let the sides opposite the corresponding vertices be the arcs , , . Let , , be the angles at the vertices , , ; , , are also the dihedral angles of a trihedron with apex and edges , , . Let , , be the angles of at . Let , , be points on the sides (or their extensions) opposite to , , . Define the unit vectors , , .
This Demonstration illustrates the following four theorems:
1. Let be the plane through and the bisector of the arc opposite ; define and similarly. Then , , meet along a common straight line , which is parallel to .
2. The bisector planes of the dihedral angles , , of the trihedron meet along a common straight line , which is parallel to .
3. The planes through the edges , , and orthogonal to the opposite faces of meet along a common straight line , which is parallel to .
4. The planes through the perpendicular bisectors of the faces of meet along a common straight line , which is parallel to .

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This Demonstration is based on problem 5.34 in [3, p. 88].
References
[1] Wikipedia. "Spherical Law of Cosines." (Mar 15, 17) en.wikipedia.org/wiki/Spherical_law_of _cosines.
[2] Wikipedia. "Spherical Trigonometry." (Mar 15, 2017) en.wikipedia.org/wiki/Spherical_trigonometry.
[3] V. V. Prasolov and I. F. Sharygin, Problems in Stereometry (in Russian), Moscow: Nauka, 1989.
[4] Wikipedia. "Ceva's Theorem." (Mar 15, 2017) en.wikipedia.org/wiki/Ceva%27 s_theorem.
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