# 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')