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

.