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 \(N\).