Question 131
Two circles $C_1$ and $C_2$, with centres at $D$ and $E$ respectively, touch at $B$. The circle having $DE$ as diameter intersects the circle $C_1$ at $H$ and the circle $C_2$ at $K$. The points $H$ and $K$ both lie on the same side of the line $DE$. $HK$ extended in both directions meets the circle $C_1$ at $L$ and meets the circle $C_2$ at $M$. Prove that
(a) $|LH| = |KM|$;
(b) the line through $B$ perpendicular to $DE$ bisects $HK$.
(a) $|LH| = |KM|$;
(b) the line through $B$ perpendicular to $DE$ bisects $HK$.