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第53页

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 （1）证明：$\because AD$平分$\angle BAC，$
$\therefore \angle BAD=\angle CAD.$

 $\because EG// AD，$$\therefore \angle BAD=\angle G，$$\angle CAD=\angle AFG.$ 

$\therefore \angle G=\angle AFG.$ $\therefore AF = AG.$ 

$\therefore \triangle AFG$是等腰三角形 

（2）解：$\because CE = EF，$$\therefore \angle CFE = \angle C.$

 $\because \angle AFG=\angle CFE，$$\angle AFG=\angle CAD，$
$\therefore \angle C=\angle CAD.$ 

$\because \angle BAC = 80^{\circ}，$$AD$平分$\angle BAC，$
$\therefore \angle C = \angle CAD = 40^{\circ}.$ 

$\therefore \angle B=180^{\circ}-\angle BAC-\angle C=60^{\circ}$

 （1）证明：$\because AB = AC，$$\angle BAC = 36^{\circ}，$
$\therefore \angle ABC = \angle ACB = \frac{1}{2}(180^{\circ}-\angle BAC)=72^{\circ}.$ 

$\because BD$是$\angle ABC$的平分线，

$\therefore \angle ABD=\frac{1}{2}\angle ABC = 36^{\circ}.$ 

$\therefore \angle BAD=\angle ABD.$ $\therefore AD = BD.$ 

又$\because E$是$AB$的中点，$\therefore DE\perp AB，$即$EF\perp AB$ 

（2）证明：$\because EF\perp AB，$$E$是$AB$的中点，

$\therefore EF$垂直平分$AB.$ $\therefore AF = BF.$ 

$\therefore \angle BAF = \angle ABF.$ 由（1），知$\angle ABD = \angle BAD = 36^{\circ}，$
$\therefore$易得$\angle FAD=\angle FBD = 72^{\circ}-36^{\circ}=36^{\circ}.$ 

又$\because \angle ACB = 72^{\circ}，$
$\therefore \angle AFC = \angle ACB-\angle CAF = 36^{\circ}.$ 

$\therefore \angle CAF=\angle AFC = 36^{\circ}.$ 

$\therefore AC = CF，$即$\triangle ACF$为等腰三角形