# Four Theorems on Spherical Triangles

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 =, =, =.

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O=(0,0,0)

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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 ++.

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2. The bisector planes of the dihedral angles , , of the trihedron meet along a common straight line , which is parallel to sin(a)+sin(b)+sin(c).

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3. The planes through the edges , , and orthogonal to the opposite faces of meet along a common straight line , which is parallel to sin(a)cos(B)cos(C)+sin(b)cos(A)cos(C)+sin(c)cos(A)cos(B).

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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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