Mollweide equations will be derived using homogenous trilinear coordinates.
In Fig. 1 is shown an arbitrary triangle ABC. Point O is the circumcenter, and C1, C2 are the first and second intersection of the circumcircle with the bisector of the side AB.
Fig. 1
Mollweide equations are as follows
(a+b) : c = cos
a-b
2
: sin
g
2
,
(1)
(a-b) : c = sin
a-b
2
: cos
g
2
.
(2)
From Fig. 1 we obtain the following trilinear coordinates of point C1
C1 = cosb+cosa : cosa+cosb : cosg-1 ,
(3)
and equivalently
C1 = sin(b+
g
2
) : sin(a+
g
2
) : -sin
g
2
.
(4)
Since b+g/2 = (b-a)/2+p/2 and a+g/2 = (a-b)/2+p/2 ,
from (4) follows
C1 = cos
b-a
2
: cos
a-b
2
: -sin
g
2
.
(5)
We recall the following trivial relations
a = b cosg+c cosb ,
(6)
b = c cosa+a cosg .
(7)
Adding (6) and (7) and after some rearrangement we obtain
(a+b)(1-cosg) = c(cosa+cosb) ,
(8)
or
(a+b) : c = cosa+cosb : 1-cosg .
(9)
Now comparing (3), (5) and (9) we obtain equation (1).
From Fig. 1 we obtain the following trilinear coordinates of point C2
C2 = cosa-cosb : cosb-cosa : 1+cosg ,
(10)
and equivalently
C2 = -cos(b+
g
2
) : -cos(a+
g
2
) : cos
g
2
.
(11)
Since b+g/2 = (b-a)/2+p/2 and a+g/2 = (a-b)/2+p/2 ,
from (11) follows
C2 = sin
b-a
2
: sin
a-b
2
: cos
g
2
.
(12)
Subtracting (7) from (6) and after some rearrangement we obtain
(a-b)(1+cosg) = c(cosb-cosa) ,
(13)
or
a-b : c = cosb-cosa : 1+cosg .
(14)
Now comparing (10), (12) and (14) we obtain equation (2).
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