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A nice property of circumscribed quadrangles

Source: German MO 2009, Round 4, Day 1, Problem 3 - 481343

May 10, 2009
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Problem Statement

Let ABCD ABCD be a (non-degenerate) quadrangle and N N the intersection of AC AC and BD BD. Denote by a,b,c,d a,b,c,d the length of the altitudes from N N to AB,BC,CD,DA AB,BC,CD,DA, respectively. Prove that \frac{1}{a}\plus{}\frac{1}{c} \equal{} \frac{1}{b}\plus{}\frac{1}{d} if ABCD 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.]
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