14.（★★★）如圖，在$\triangle ABC$中，$BD$平分$\angle ABC$，$BD\perp AC$于點$D$，點$F$在$AC$上，
點$E$在$AB$的延長線上，連接$EF$交$BC$于點$G$，且$\angle ABC = 2\angle E$。
(1)試問：$EF$與$AC$有怎樣的位置關系？請說明理由.
(2)若$\angle E = 40^{\circ}$，請求出$\angle C$的度數。
答案：(1) $EF\perp AC$. 理由如下： $BD$ 平分 $\angle ABC$，
$\therefore\angle ABC = 2\angle ABD$.
$\because\angle ABC = 2\angle E$,
$\therefore\angle ABD=\angle E$. $\therefore BD// EF$.
$\therefore\angle ADB=\angle AFE$.
$\because BD\perp AC$,
$\therefore\angle ADB = 90^{\circ}$. $\therefore\angle AFE = 90^{\circ}$.
$\therefore EF\perp AC$.
(2) $\because\angle E = 40^{\circ}$, $\angle ABC = 2\angle E$,
$\therefore\angle ABC = 80^{\circ}$.
$\because BD// EF$,
$\therefore\angle BCE=\angle CBD=\frac{1}{2}\angle ABC=\angle E = 40^{\circ}$.
$\because\angle CGF = 40^{\circ}$.
由(1)可知，$\angle AFE = 90^{\circ}$,
$\therefore\angle GFC = 90^{\circ}$.
$\therefore\angle C = 90^{\circ}-\angle CGF = 50^{\circ}$.
