National problem · Geometry

A parallelogram \(ABCD\) satisfies \(|AB| = |BD|\). Let \(K\) be the point of line \(AB\), different from \(A\), with \(|KD| = |AD|\). Let \(M\) be the image of \(C\) under the half-turn about \(K\) (so \(K\) is the midpoint of \(CM\)), and let \(N\) be the image of \(B\) under the half-turn about \(A\) (so \(A\) is the midpoint of \(BN\)).

Prove that the triangle \(MDN\) is isosceles with apex at \(D\).

Single ticks mark \(|NA| = |AB| = |BD|\), double ticks mark \(|AD| = |DK|\). The half-turn about \(K\) sends \(C\) to \(M\), the half-turn about \(A\) sends \(B\) to \(N\).

Slovenian High School Mathematics Competition for Vega Awards (MaSSA), drzavno (national) round 2014, 1. letnik, category A, problem 3. Organized by DMFA Slovenije (Society of Mathematicians, Physicists and Astronomers of Slovenia). Source

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