# 直角三角形

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## 主要性質

### 面積

${\displaystyle T={\tfrac {1}{2}}ab}$

${\displaystyle T={\text{PA}}\cdot {\text{PB}}=(s-a)(s-b).}$

### 高

• 高為斜線切割出的二線段的幾何平均數
• 各股是直角三角形的高和斜線切割出的二線段中相鄰部份的幾何平均數。

${\displaystyle \displaystyle f^{2}=de,}$（有時稱為直角三角形高定理
${\displaystyle \displaystyle b^{2}=ce,}$
${\displaystyle \displaystyle a^{2}=cd}$

${\displaystyle f={\frac {ab}{c}}.}$

${\displaystyle {\frac {1}{a^{2}}}+{\frac {1}{b^{2}}}={\frac {1}{f^{2}}}.}$

### 勾股定理

${\displaystyle \displaystyle a^{2}+b^{2}=c^{2}}$

### 內切圓及外接圓

${\displaystyle r={\frac {a+b-c}{2}}={\frac {ab}{a+b+c}}.}$

${\displaystyle R={\frac {c}{2}}.}$

${\displaystyle \displaystyle a={\frac {2r(b-r)}{b-2r}}.}$

## 性質

### 邊長和半周長

• ${\displaystyle \displaystyle a^{2}+b^{2}=c^{2}}$勾股定理
• ${\displaystyle \displaystyle (s-a)(s-b)=s(s-c)}$
• ${\displaystyle \displaystyle s=2R+r.}$[7]
• ${\displaystyle \displaystyle a^{2}+b^{2}+c^{2}=8R^{2}.}$[8]

### 角

• A和角B互為餘角
• ${\displaystyle \displaystyle \cos {A}\cos {B}\cos {C}=0.}$[8][9]
• ${\displaystyle \displaystyle \sin ^{2}{A}+\sin ^{2}{B}+\sin ^{2}{C}=2.}$[8][9]
• ${\displaystyle \displaystyle \cos ^{2}{A}+\cos ^{2}{B}+\cos ^{2}{C}=1.}$[9]
• ${\displaystyle \displaystyle \sin {2A}=\sin {2B}=2\sin {A}\sin {B}.}$

### 面積

• ${\displaystyle \displaystyle T={\frac {ab}{2}}}$
• ${\displaystyle \displaystyle T=r_{a}r_{b}=rr_{c}}$
• ${\displaystyle \displaystyle T=r(2R+r)}$
• ${\displaystyle T=PA\cdot PB,}$其中P為內切圓和最長邊AB相切的點[10]

### 內切圓及外切圓半徑

• ${\displaystyle \displaystyle r=s-c}$[11]
• ${\displaystyle \displaystyle r_{a}=s-b}$[11]
• ${\displaystyle \displaystyle r_{b}=s-a}$[11]
• ${\displaystyle \displaystyle r_{c}=s}$[11]
• ${\displaystyle \displaystyle r_{a}+r_{b}+r_{c}+r=a+b+c}$[11]
• ${\displaystyle \displaystyle r_{a}^{2}+r_{b}^{2}+r_{c}^{2}+r^{2}=a^{2}+b^{2}+c^{2}}$[11]
• ${\displaystyle \displaystyle r={\frac {r_{a}r_{b}}{r_{c}}}}$[11]

### 高線和中線

• ${\displaystyle \displaystyle h={\frac {ab}{c}}}$
• ${\displaystyle \displaystyle m_{a}^{2}+m_{b}^{2}+m_{c}^{2}=6R^{2}.}$[12]
• 中線中有一條的長度等於外接圓半徑。
• 高線中最短的（通過由最大角頂點的高線）將對邊分為二個線段，高線恰為二線段的幾何平均數，即為直角三角形高定理

### 內切參圓和外接圓

• 三角形可以放在一個半圓英語Semicircle中，且一邊恰和直徑完全重合。
• 外接圓圓心恰為最長邊的中點。
• 最長邊的邊長恰為外接圓的直徑。${\displaystyle \displaystyle (c=2R).}$
• 外接圓和九點圓相切[8]
• 垂心在外接圓的圓周上[12]
• 內切圓圓心（內心）和垂心的距離為${\displaystyle {\sqrt {2}}r}$.[12]

## 各邊的比例

a, b, h為角A的對邊、鄰邊和斜邊

${\displaystyle \sin \theta ={\frac {a}{h}},\,\cos \theta ={\frac {b}{h}},\,\tan \theta ={\frac {a}{b}},\,\sec \theta ={\frac {h}{b}},\,\cot \theta ={\frac {b}{a}},\,\csc \theta ={\frac {h}{a}}.}$

## 中線

${\displaystyle m_{a}^{2}+m_{b}^{2}=5m_{c}^{2}={\frac {5}{4}}c^{2}.}$

## 不同平均和黃金比例的關係

HGA是二個正整數aba > b）的調和平均幾何平均算術平均。若一直角三角形的二股為HG，其斜邊為A，則[13]

${\displaystyle {\frac {A}{H}}={\frac {A^{2}}{G^{2}}}={\frac {G^{2}}{H^{2}}}=\phi \,}$

${\displaystyle {\frac {a}{b}}=\phi ^{3},\,}$

## 其他性質

${\displaystyle p^{2}+q^{2}=5\left({\frac {c}{3}}\right)^{2}}$

hkh > k）為一斜邊長為c的直角三角形的二個內接正方形邊長，則

${\displaystyle {\frac {1}{c^{2}}}+{\frac {1}{h^{2}}}={\frac {1}{k^{2}}}.}$

## 參考資料

1. ^ 勾股弦概說. 科博館. [2013-08-22].
2. ^ A. Aleksei Petrovich Stakhov. Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer. World Scientific. 2009: p.86. ISBN 9812775838.
3. ^ Di Domenico, Angelo S., "A property of triangles involving area", Mathematical Gazette 87, July 2003, pp. 323-324.
4. ^ Wentworth, G.A. A Text-Book of Geometry. Ginn & Co. 1895.
5. ^ Voles, Roger, "Integer solutions of ${\displaystyle a^{-2}+b^{-2}=d^{-2}}$," Mathematical Gazette 83, July 1999, 269–271.
6. ^ Richinick, Jennifer, "The upside-down Pythagorean Theorem," Mathematical Gazette 92, July 2008, 313–317.
7. ^ Triangle right iff s = 2R + r. Art of problem solving. 2011-06-11 [2013-08-24].
8. Andreescu, Titu and Andrica, Dorian, "Complex Numbers from A to...Z", Birkhäuser, 2006, pp. 109-110.
9. A Variant of the Pythagorean Theorem. CTK Wiki Math. 2012-10-17 [2013-08-24].
10. ^ Darvasi, Gyula, Converse of a Property of Right Triangles, The Mathematical Gazette, March 2005, 89 (514): 72–76.
11. Bell, Amy, Hansen's Right Triangle Theorem, Its Converse and a Generalization (PDF), Forum Geometricorum, 2006, 6: 335–342.
12. Inequalities proposed in 「Crux Mathematicorum」, Problem 954, p. 26, .
13. ^ Di Domenico, A., "The golden ratio — the right triangle — and the arithmetic, geometric, and harmonic means," Mathematical Gazette 89, July 2005, 261. Also Mitchell, Douglas W., "Feedback on 89.41", vol 90, March 2006, 153-154.
14. ^ Posamentier, Alfred S., and Salkind, Charles T. Challenging Problems in Geometry, Dover, 1996.
15. ^ Bailey, Herbert, and DeTemple, Duane, "Squares inscribed in angles and triangles", Mathematics Magazine 71(4), 1998, 278-284.