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}\plus{}\frac{1}{c} \equal{} \frac{1}{b}\plus{}\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.] searchgeometry proposedgeometry