Question

The line segment s bisects the angle C so that the two angles labeled x are equal. Show that the length of this angle bisector is s= 2abcosx/a + b

Answer 1

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)