# 射影定理

## 定理内容

ΔABC 中，C = 90°，以及 CDABADBD 分别是 ACBC 在底边 AB 的正投影。

ΔABC 中，C = 90°。设 CDAB 的上的高，则有：

${\displaystyle {AC}^{2}=AD\cdot AB}$
${\displaystyle {BC}^{2}=BD\cdot AB}$

## 证明

${\displaystyle {\frac {AB}{AC}}={\frac {AC}{AD}}}$

${\displaystyle {AC}^{2}=AD\cdot AB}$

${\displaystyle {\frac {AB}{BC}}={\frac {BC}{BD}}}$

${\displaystyle {BC}^{2}=BD\cdot AB}$

## 相关定理

### 直角三角形面积

${\displaystyle AB\cdot CD=AC\cdot BC}$

### 勾股定理

${\displaystyle {AB}^{2}={AC}^{2}+{BC}^{2}}$

${\displaystyle {AC}^{2}=AD\cdot AB}$
${\displaystyle {BC}^{2}=BD\cdot AB}$

${\displaystyle {AC}^{2}+{BC}^{2}=AD\cdot AB+BD\cdot AB}$

${\displaystyle {AB}^{2}={AC}^{2}+{BC}^{2}}$

### 几何平均定理

${\displaystyle {CD}^{2}=AD\cdot BD}$

## 一般三角形的情况

${\displaystyle AD=AC\cos \angle A}$
${\displaystyle BD=BC\cos \angle B}$

${\displaystyle AB=AC\cos \angle A+BC\cos \angle B}$

## 三维空间上的推广

### 三直角四面体

${\displaystyle [\triangle ADB]^{2}=[\triangle AEB]\cdot [\triangle ABC]}$
${\displaystyle [\triangle ADC]^{2}=[\triangle AEC]\cdot [\triangle ABC]}$
${\displaystyle [\triangle BDC]^{2}=[\triangle BEC]\cdot [\triangle ABC]}$

${\displaystyle [\triangle ABC]^{2}=[\triangle ADB]^{2}+[\triangle ADC]^{2}+[\triangle BDC]^{2}}$

### 一般四面体

${\displaystyle AE=AD\cos \alpha }$
${\displaystyle BE=BD\cos \beta }$
${\displaystyle CE=CD\cos \gamma }$

${\displaystyle [\triangle ABE]=[\triangle ABD]\cos \theta }$
${\displaystyle [\triangle ACE]=[\triangle ACD]\cos \phi }$
${\displaystyle [\triangle BCE]=[\triangle BCD]\cos \psi }$

${\displaystyle [\triangle ABC]=[\triangle ABD]\cos \theta +[\triangle ACD]\cos \phi +[\triangle BCD]\cos \psi }$

### 任意图形的投影

${\displaystyle S_{\mathrm {proj} }=S\cos \theta }$