Problem 3, 2018

Let \(ABC\) be a right triangle with hypotenuse \(AB\), and let \(D\) be the midpoint of \(AB\). Let \(k\) be the circumcircle of the triangle \(BCD\), and let \(E\) be an arbitrary point of the shorter arc \(BD\) of \(k\). On the line \(BC\), a point \(F\) is taken so that \(B\) lies between \(C\) and \(F\) and

\(\angle BEF = 2\angle BAF .\)

Let \(k_1\) be the circumcircle of the triangle \(CEF\). Prove that one of the common tangents of the circles \(k\) and \(k_1\) passes through the point \(D\).