NL-NWO 1963 Problem 2

The straight lines $k$ and $\ell$ intersect at right angles. A line intersects $k$ in $A$ and $\ell$ in $B$ . Consider all straight line segments $PQ$ ( $P$ on $k$ and $Q$ on $\ell$ ), which makes an angle of $45^o$ with $AB$ . (a) Determine the locus of the midpoints of the line segments $PQ$ , (b) If the perpendicular bisector of such a line segment $PQ$ intersects the line $k$ at $K$ and the line $\ell$ at $L$ , then prove that $KL \ge PQ$ .