Austrian Mathematical Olympiad

Figure 3: Problem 14

Let $t_1$ be the tangent line to $k_1$ in point $E$ and let $t_2$ be the tangent line to $k_2$ in point $E$. The tangent-secant theorem applied to circle $k_1$ gives $$\angle (EF,t_1)=\angle FAE=\alpha$$ with the usual notation for the angles in triangle $ABC$. The tangent-secant theorem applied to circle $k_2$ gives $$\angle (EF,t_2)=\angle SDE=\angle CDE=\alpha ,$$ where the last equality comes from the fact that $ABDE$ is a cyclic quadrilateral since all four vertices lie on the Thales circle with diameter $AB$. Therefore, $t_1$ and $t_2$ are parallel and they both contain the point $E$. So, the two tangents are identical which implies that the circles touch in $E$.