Consider a homothety with center \(O\) and coefficient \(k\). Which lines are sent to themselves by this homothety? (Hint: the answer will depend on \(k\))
In a triangle \(ABC\) we have \(AB = AC\). A circle which is internally tangent to the circumcircle of the triangle is also tangent to the sides \(AB\) and \(AC\) at the points \(P\) and \(Q\), respectively. Prove that the midpoint of \(PQ\) is the centre of the incircle of triangle \(ABC\).