Question 23
A circle $\omega_1$ centred at $O_1$ intersects another circle $\omega_2$ centred at $O_2$ at two distinct points $P$ and $Q$. Points $A$ and $B$ are on $\omega_1$ and $\omega_2$ respectively such that $AB$ is an external common tangent to $\omega_1$ and $\omega_2$. The line through $PQ$ intersects the segments $AB$ and $O_1O_2$ at $M$ and $N$ respectively. Suppose the radius of $\omega_1$ is 143 cm, the radius of $\omega_2$ is 78 cm and $O_1O_2=169$ cm. Determine the length of $MN$ in centimetres.