DE-MO 2009 Problem 3

Let $ABCD$ be a (non-degenerate) quadrangle and $N$ the intersection of $AC$ and $BD$ . Denote by $a,b,c,d$ the length of the altitudes from $N$ to $AB,BC,CD,DA$ , respectively.

Prove that $\frac{1}{a}+\frac{1}{c} = \frac{1}{b}+\frac{1}{d}$ if $ABCD$ has an incircle.

Extension: Prove that the converse is true, too.

[If this has already been posted, I humbly apologize. A quick search turned up nothing.]