答案：
【详解】 (1) 解: $\mathrm{Q} A E \perp B D$,
$$
\begin{aligned}
& \therefore \angle A E O=90^{\circ}, \\
& \because \angle A O E=50^{\circ}, \\
& \therefore \angle E A O=40^{\circ}, \\
& \mathrm{Q} C A \text { 平分 } \angle D A E, \\
& \therefore \angle D A C=\angle E A O=40^{\circ},
\end{aligned}
$$
$\mathrm{Q}$ 四边形 $A B C D$ 是平行四边形,
$$
\begin{aligned}
& \therefore A D / / B C, \\
& \angle A C B=\angle D A C=40^{\circ},
\end{aligned}
$$
(2) 证明: $\mathrm{Q}$ 四边形 $A B C D$ 是平行四边形,
$$
\begin{aligned}
& \therefore O A=O C, \\
& \mathrm{Q} A E \perp B D, C F \perp B D, \\
& \therefore \angle A E O=\angle C F O=90^{\circ}, \\
& \mathrm{Q} \angle A O E=\angle C O F,
\end{aligned}
$$
$$
\begin{aligned}
& \therefore \triangle A E O \cong \triangle C F O(A A S), \\
& \therefore A E=C F .
\end{aligned}
$$