Geometry · Circles · Tangent lines · Thales theorem · Angle chasing
A circle \(k\) has diameter \(AB\). A point \(M\) of \(k\) is chosen, different from \(A\) and from \(B\). Let \(k_1\) be the circle with centre \(M\) that touches the diameter \(AB\).
The line \(AB\) is itself a tangent to \(k_1\), so each of the points \(A\) and \(B\) sends a second tangent to \(k_1\). Prove that these two tangents - the one from \(A\) and the one from \(B\) - are parallel.
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Slovenian High School Mathematics Competition for Vega Awards (MaSSA), drzavno (national) round 2003, 1. letnik, category A, problem 3. Organized by DMFA Slovenije (Society of Mathematicians, Physicists and Astronomers of Slovenia). Source