1.如图，△ABC中，BO平分∠ABC，CO平分∠ACB，MN经过点O，且MN//BC分别交AB，AC于M，N.若AB＝12，AC＝18，直接写出△AMN的周长为.

30

2.如图，在△ABC中，∠ACB=90°，点D在AB上，AD=CD，求证：AD=BD.

证明:$\because AD=CD$,
$\therefore ∠ A=∠ ACD$,
又$\because ∠ ACB=90°$,
$\therefore ∠ B=90°-∠ A$,
$∠ DCB=90°-∠ ACD$,
$\therefore ∠ B=∠ DCB$,
$\therefore CD=BD$,
$\therefore AD=BD$.

3.如图,在$△ ABC$中,$∠ C=3∠ B$,$AD$平分$∠ BAC$交$BC$于$D$点,$DE⊥ AD$交$AB$于点$E$,求证:$BE=DE$.

证明:设$∠ B=α$,$∠ C=3α$,
$\therefore ∠ DAE=\dfrac{180°-4α}{2}=90°-2α$,
$\therefore ∠ DEA=2α=∠ B+∠ BDE$,
$∠ BDE=α=∠ B$,
$\therefore DE=BE$.

4.如图,在△ABC中,CE为角平分线,AD⊥CE于点F,交BC于点D,AG//BC交CE的延长线于G点,∠ACB=2∠B.
(1)求证:BE=CE;
(2)求证:AB=2CF.

4.证明:(1)设$∠ B=α$,则$∠ ACB=2α$,
$\therefore ∠ BCE=α$,$\therefore ∠ BCE=∠ B$,
$\therefore BE=CE$;
(2)$\because AG// BC$,
$\therefore ∠ G=∠ GCB=∠ B=α$,
$\therefore ∠ G=∠ ACG$,
$\therefore AG=AC$,
又$\because AF⊥ CE$,$\therefore FG=CF$,$\therefore CG=2CF$,
而$∠ G=∠ B=∠ EAG$,
$\therefore AE=EG$,
$\therefore AB=CG=2CF$.

5.如图，在△ABE中，AB＝AE，AC⊥BE于点C，D在BC上，∠ABC＝2∠CAD，AB＝10，BC＝8，求CD的长。

解:设$∠ CAD=α$,
$∠ ADC=90°-α$,
$∠ EAD=90°-α$,
$\therefore AB=AE=ED$,
$\therefore BD=2BC-AB$,
$\therefore BD=16-10=6$,
$\therefore CD=2$.