Geometry · Tangent circles · Pythagorean theorem · Isosceles triangles · Angle chasing · Parallel lines
Three curves are drawn inside a square \(ABCD\): the quarter circle \(\mathcal{Q}\) centred at the vertex \(A\) and passing through \(B\) and \(D\); the semicircle \(\mathcal{P}\) centred at the midpoint of the side \(AD\) and passing through \(A\) and \(D\); and the circle \(\mathcal{K}\), which touches \(\mathcal{P}\) at the point \(E\), touches \(\mathcal{Q}\) at the point \(F\), and touches the side \(AB\) at the point \(G\).
(a) Express the radius of \(\mathcal{K}\) in terms of the side length of the square \(ABCD\).
(b) Prove that the line \(EF\) is parallel to the side \(AB\).
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Slovenian High School Mathematics Competition for Vega Awards (MaSSA), drzavno (national) round 2022, 1. letnik, category A, problem B2. Organized by DMFA Slovenije (Society of Mathematicians, Physicists and Astronomers of Slovenia). Source