Geometry · Isosceles right triangles · Parallelograms · Midlines · Invariance · Auxiliary points
Let \(AB\) be a segment and let \(P\) be any point of it other than \(A\) and \(B\). On the hypotenuses \(AP\) and \(PB\) erect isosceles right triangles \(APQ\) and \(PBR\), with the right angles at the apexes \(Q\) and \(R\), so that \(Q\) and \(R\) lie on the same side of the line \(AB\). Let \(M\) be the midpoint of the segment \(QR\).
Prove that the distance from \(M\) to the line \(AB\) does not depend on the choice of \(P\).
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Slovenian High School Mathematics Competition for Vega Awards (MaSSA), drzavno (national) round 2001, 1. letnik, category A, problem 3. Organized by DMFA Slovenije (Society of Mathematicians, Physicists and Astronomers of Slovenia). Source