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WOLFRAM|DEMONSTRATIONS PROJECT

Four Theorems on Spherical Triangles

​
theorem
1
2
3
4
point A
1.0472
point B
1.0472
1.08331
additional rays
show sphere
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
. Let
α
,
β
,
γ
be the angles at the vertices
A
,
B
,
C
;
α
,
β
,
γ
are also the dihedral angles of a trihedron
τ
with apex
O
and edges
OA
,
OB
,
OC
. Let
a
,
b
,
c
be the angles of
τ
at
O
. Let
A'
,
B'
,
C'
be points on the sides (or their extensions) opposite to
A
,
B
,
C
. Define the unit vectors
u
A
=
OA
,
u
B
=
OB
,
u
C
=
OC
.
This Demonstration illustrates the following four theorems:
1. Let
π
A
be the plane through
OA
and the bisector of the arc
a
opposite
A
; define
π
B
and
π
C
similarly. Then
π
A
,
π
B
,
π
C
meet along a common straight line
OR
, which is parallel to
u
A
+
u
B
+
u
C
.
2. The bisector planes of the dihedral angles
α
,
β
,
γ
of the trihedron meet along a common straight line
OS
, which is parallel to
u
A
sin(a)+
u
B
sin(b)+
u
C
sin(c)
.
3. The planes through the edges
a
,
b
,
c
and orthogonal to the opposite faces of
τ
meet along a common straight line
OT
, which is parallel to
u
A
sin(a)cos(B)cos(C)+
u
B
sin(b)cos(A)cos(C)+
u
C
sin(c)cos(A)cos(B)
.
4. The planes through the perpendicular bisectors of the faces of
τ
meet along a common straight line
OU
, which is parallel to
u
A
u
B
+
u
B
u
C
+
u
C
u
A
.
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