查看源代码：
geometry-of-mirror.md

---
title: 平面镜成像性质的数学原理
createTime: 2022/11/10
categories:
    - study
tags:
    - maths
    - physics
---

~~千万不要将此博文发给你的数学和物理老师~~

![](image/1668126125168.png)

$\because GP\perp PQ$  
$\therefore \angle APG=\angle GPM=90\degree$  
$\text{又}\because \angle 1+\angle 2=\angle APG, \angle 3+\angle SPQ=\angle GPM$  
$\therefore \angle 1=\angle SPQ$  
$\text{又}\because AB, S'P\text{交于点}P$  
$\therefore \angle 1=\angle S'PQ$  
$\therefore \angle SPQ=\angle S'PQ$  
$\text{同理，}\angle SQP=\angle S'QP$  

$\because \text{在}\triangle SPQ\text{和}\triangle S'PQ\text{中}$  
$\begin{cases} \angle SPQ=\angle S'PQ \\ PQ=PQ \\ \angle SQP=\angle S'QP\end{cases}$  
$\triangle SPQ≌\triangle S'PQ(ASA)$<!--LaTeX中的全等符号是反的，故直接打出-->  
$\therefore SP=S'P$  

$\because \text{在}\triangle SPM\text{和}\triangle S'PM\text{中}$  
$\begin{cases} SP=S'P \\ \angle SPM=\angle S'PM \\ PM=PM\end{cases}$  
$\triangle SPM≌\triangle S'PM(SAS)$  
$\therefore SM=S'M, \angle SMP=\angle S'MP$  

$\text{又}\because M\text{在}SS'\text{上}$  
$\therefore \angle SMP+\angle S'MP=180\degree$  
$\therefore \angle SMP=\angle S'MP=180\degree \div 2=90\degree$  
$\therefore AB \perp SS'$