Menelaus' and Ceva's Theorem for Spherical Triangle
Menelaus' and Ceva's Theorem for Spherical Triangle
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:
ABC
O=(0,0,0)
a
b
c
A'
B'
C'
The rays , , are on the same plane if and only if
OA'
OB'
OC'
sin(OA,OC')sin(OB,OA')sin(OC,OB')=sin(OB,OC')sin(OC,OA')sin(OA,OB')
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
(OA,OA')
(OB,OB')
(OC,OC')
OS
sin(OA,OC')sin(OB,OA')sin(OC,OB')=sin(OB,OC')sin(OC,OA')sin(OA,OB')