A geometry problem by Suresh Bala

All of the brown angles $\theta$ are the same size, since $\angle ACE$ is bisected by $DC$ . Also both green angles $\alpha$ are the same, since $ABCD$ is inscribed in a circle.

Law of sines in $\triangle ACD$ can be written as $\dfrac{AD}{\sin(180^\circ-\theta)}=\dfrac{DC}{\sin\alpha}$

From this $AD=8=\dfrac{DC\sin\theta}{\sin\alpha}$

Similarly in $\triangle BCD$ the law of sines is $\dfrac{BD}{\sin\theta}=\dfrac{DC}{\sin\alpha}$

And from this $BD=\dfrac{DC\sin\theta}{\sin\alpha}=\boxed8$