ILL 1985 Problem 46

Let $C$ be the curve determined by the equation $y = x^3$ in the rectangular coordinate system. Let $t$ be the tangent to $C$ at a point $P$ of $C$ ; t intersects $C$ at another point $Q$ . Find the equation of the set $L$ of the midpoints $M$ of $PQ$ as $P$ describes $C$ . Is the correspondence associating $P$ and $M$ a bijection of $C$ on $L$ ? Find a similarity that transforms $C$ into $L.$