Problem 12.4. Let $A B C D$ be a circumscribed quadrilateral. Find $\Varangle B C D$ if $A C=B C, A D=5, E=A C \cap B D, B E=12$ and $D E=3$.

Oleg Mushkarov, Nikolai Nikolov
12.4. If the perpendicular bisector of $C D$ meets $B D$ at point $O$, then

$$
\begin{aligned}
\Varangle C O D & =180^{\circ}-2 \Varangle O D C \\
& =180^{\circ}-2 \Varangle B A C \\
& =\Varangle A C B=\Varangle A D O
\end{aligned}
$$

and therefore $A D \| C O$. Hence $\frac{O E}{3}=\frac{C O}{A D}=$ $\frac{O E+3}{5}$, which implies that $O E=\frac{9}{2}$. This shows that $O$ is the midpoint of $B D$ and then

![](https://cdn.mathpix.com/cropped/2024_06_03_f8ccd71b4db367e7c2a9g-109.jpg?height=567&width=507&top_left_y=1295&top_left_x=1159)
$\Varangle B C D=90^{\circ}$.