Verifying parts of a triangle

Question

Here is the triangle

S bisects angle C.

What is S's length?

Basically, how do I show that S = (2abcosx)/(a+b)

I was told using Law of Sines is a hint. The professor asked me to do this as a verification problem / proof. Any help would be appreciated.

Answer 1

Use the rule for areas.

The two smaller triangles have areas $\frac 12 aS \sin x$ and $\frac 12 bS \sin x$.

The whole triangle has area $\frac 12 ab \sin (2x)$

The area of the one is the sum of the areas of the two smaller:

$\frac 12 ab \sin (2x) = \frac 12 aS \sin x + \frac 12 bS \sin x$

$ab \sin (2x) = aS \sin x + bS \sin x$

$2ab \sin x \cos x = (a+b)S \sin x$

... and there you have it.