# 平分線

## 角平分线的性质

${\displaystyle OM}$ 平分${\displaystyle {\angle }AOB,P}$ ${\displaystyle OM}$ 上一点${\displaystyle ,PE{\perp }OA}$ ${\displaystyle E,PF{\perp }OB}$ ${\displaystyle F,}$

${\displaystyle PE=PF.}$

### 该性质的证明

${\displaystyle {\because \quad }OM}$ 平分${\displaystyle {\angle }AOB,}$

${\displaystyle {\therefore \quad \angle }POE={\angle }POF.}$

${\displaystyle {\because \quad }PE{\perp }OA,\;PF{\perp }OB,}$

${\displaystyle {\therefore \quad \angle }OEP={\angle }OFP=90^{\circ }.}$

${\displaystyle {\triangle }OEP}$ ${\displaystyle {\triangle }OFP}$ ${\displaystyle ,}$

${\displaystyle {\begin{cases}{\angle }OEP={\angle }OFP,\\{\angle }POE={\angle }POF,\\OP=OP,\end{cases}}}$

${\displaystyle {\therefore \quad \triangle }OEP{\;\cong \triangle }OFP({\mbox{AAS}}).}$

${\displaystyle {\therefore \quad }PE=PF.}$

## 角平分线的判定

### 证明

${\displaystyle {\because \quad }PE{\perp }OA,\;PF{\perp }OB,}$

${\displaystyle {\therefore \quad \angle }OEP={\angle }OFP=90^{\circ }.}$

${\displaystyle \mathrm {Rt} {\triangle }OEP}$ ${\displaystyle \mathrm {Rt} {\triangle }OFP}$ ${\displaystyle ,}$

${\displaystyle {\begin{cases}OP=OP,\\PE=PF,\end{cases}}}$

${\displaystyle {\therefore \quad }\mathrm {Rt} {\triangle }OEP\;{\cong }\mathrm {Rt} {\triangle }OFP(\mathrm {HL} ).}$

${\displaystyle {\therefore \quad \angle }POE={\angle }POF,}$

${\displaystyle {\therefore \quad }OM}$ 平分${\displaystyle {\angle }AOB.}$