If we have a triangle with sides a and b and an included angle of C, then the area of the triangle would be:
A = (1/2) ab sin C
If angle C is bisected into two each angles each measuring x, then the area can be expressed as:
A = (1/2) ab sin 2x
Using the trigonometric identity for sin 2x = 2 sin x cos x, the area would now be:
A = ab sin x cos x
Since the line segment s divides the angle into two, it also divides the triangle into two. Another equation for the area is:
A = (1/2) as sin x + (1/2) bs sin x
Equating the two equations gives us:
ab cos x = (1/2) as + (1/2) bs
Solving for s
s = 2 ab cos x / (a + b)