VMO 1969 Problem 4

Two circles centers $O$ and $O'$ , radii $R$ and $R'$ , meet at two points. A variable line $L$ meets the circles at $A, C, B, D$ in that order and $\frac{AC}{AD} = \frac{CB}{BD}$ . The perpendiculars from $O$ and $O'$ to $L$ have feet $H$ and $H'$ . Find the locus of $H$ and $H'$ . If $OO'^2 < R^2 + R'^2$ , find a point $P$ on $L$ such that $PO + PO'$ has the smallest possible value. Show that this value does not depend on the position of $L$ . Comment on the case $OO'^2 > R^2 + R'^2$ .