Proving $ \frac{1}{c} = \frac{1}{a} + \frac{1}{b}$ in a geometric context 
Prove or disprove
$$
\frac{1}{c} = \frac{1}{a} + \frac{1}{b}.
$$
I have no idea where to start, but it must be a simple proof.
Trivia. This fact was used for determination of resistance of two parallel resistors in some circumstances long time ago.

Consider the areas of triangles. Let $[ABC]$ be the area of a triangle $ABC$.
Since $[OAB]=[OAC]+[OBC]$, one has
$$\frac{1}{2}ab\sin120^\circ=\frac{1}{2}ac\sin60^\circ+\frac 12bc\sin60^\circ.$$
This leads
$$ab=ac+bc.$$
Now divide the both sides by $abc$.
