Menelaus' and Ceva's Theorem for Spherical Triangle

​
Menelaus
Ceva
point A
point B
spherical θ
spherical ϕ
point A'
point B'
additional rays
show sphere
show sides' labels
sin(OA, OC')
sin(OB, OA')
sin(OC, OB')
product
0.615289
0.59604
0.519861
0.190652
sin(OB, OC')
sin(OC, OA')
sin(OA, OB')
product
0.999845
0.39736
0.479872
0.190652
Draw a spherical triangle
ABC
on the surface of a unit sphere centered at
O=(0,0,0)
. Let the sides opposite the corresponding vertices be the arcs
a
,
b
,
c
and contain the points
A'
,
B'
,
C'
. Menelaus's theorem for a spherical triangle states:
The rays
OA'
,
OB'
,
OC'
are on the same plane if and only if
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
(OA,OA')
,
(OB,OB')
and
(OC,OC')
go through the same ray (
OS
) if and only if
sin(OA,OC')sin(OB,OA')sin(OC,OB')=sin(OB,OC')sin(OC,OA')sin(OA,OB')
.

Details

A proof of these theorems can be found in[3, pp. 85–86].

References

[1] Wikipedia. "Spherical Law of Cosines." (Mar 20, 2017) en.wikipedia.org/wiki/Spherical_law_of _cosines.
[2] Wikipedia. "Spherical Trigonometry." (Mar 20, 2017) en.wikipedia.org/wiki/Spherical_trigonometry.
[3] V. V. Prasolov and I. F. Sharygin, Problems in Stereometry (in Russian), Moscow: Nauka, 1989.

External Links

Spherical Trigonometry (Wolfram MathWorld)
Spherical Triangle Solutions
Menelaus' Theorem
Menelaus' Theorem (Wolfram MathWorld)
Ceva's Theorem (Wolfram MathWorld)

Permanent Citation

Izidor Hafner
​
​"Menelaus' and Ceva's Theorem for Spherical Triangle"​
​http://demonstrations.wolfram.com/MenelausAndCevasTheoremForSphericalTriangle/​
​Wolfram Demonstrations Project​
​Published: March 22, 2017