# 平分線

## 角平分線的性質

### 性質

${\displaystyle OM}$平分${\displaystyle {\angle }AOB,P}$${\displaystyle OM}$上一點${\displaystyle ,PE{\perp }OA}$${\displaystyle E,PF{\perp }OB}$${\displaystyle F,}$

${\displaystyle PE=PF.}$

### 該性質的證明

${\displaystyle {\because \quad }OM}$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 ,}$

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

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

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

## 角平分線的判定

### 證明

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

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

${\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${\displaystyle {\angle }AOB.}$