Problem 2, 2017

Let \(k\) be a circle with centre \(O\), and let \(T\) be a point outside it. The two tangents drawn from \(T\) touch \(k\) at the points \(A\) and \(B\). Let \(k'\) be the circle with centre \(T\) that passes through \(A\) and \(B\).

Let \(C\) be a point of \(k'\) lying outside \(k\), such that the line \(CA\) meets \(k\) once more at \(D\) and the line \(CB\) meets \(k\) once more at \(E\), the points lying in the orders \(C - A - D\) and \(C - E - B\).

Prove that \(DE\) is a diameter of the circle \(k\).