This is the first problem of #PeruMOTraining, you can see my first post here. I proposed this problem for the Peruvian Mathematical Olympiad, in 2011. Please post your solutions!
Problem Let \(ABC\) be a right triangle, with \(\angle ABC=90^\circ\). Let \(CM\) and \(AN\) be interior bisectors intersecting at \(I\) (\(M\) is on the segment \(AB\) and \(N\) is on the segment \(BC\) ). Construct the paralellograms \(AMIP\) and \(CNIQ\). If \(U\) and \(V\) are the midpoints of segments \(AC\) and \(PQ\), respectively. Prove that \(UV\) and \(AC\) are perpendicular.