Question 94
Pentagon $A B C D E$ is given with the following conditions:
(a) $\angle C B D + \angle D A E = \angle B A D = 45^{\circ}, \angle B C D + \angle D E A = 300^{\circ}$
(b) $\frac{B A}{D A} = \frac{2 \sqrt{2}}{3},\ C D = \frac{7 \sqrt{5}}{3}$, and $D E = \frac{15 \sqrt{2}}{4}$
(c) $A D^{2} \cdot B C = A B \cdot A E \cdot B D$
Compute $B D$.
Supplemental illustration (AI-generated; triangle ABD only):
(a) $\angle C B D + \angle D A E = \angle B A D = 45^{\circ}, \angle B C D + \angle D E A = 300^{\circ}$
(b) $\frac{B A}{D A} = \frac{2 \sqrt{2}}{3},\ C D = \frac{7 \sqrt{5}}{3}$, and $D E = \frac{15 \sqrt{2}}{4}$
(c) $A D^{2} \cdot B C = A B \cdot A E \cdot B D$
Compute $B D$.
Supplemental illustration (AI-generated; triangle ABD only):