# 特殊直角三角形

（重定向自等腰直角三角形

## 各角有特殊關係

0 0 ${\displaystyle {\tfrac {\sqrt {0}}{2}}=0}$ ${\displaystyle {\tfrac {\sqrt {4}}{2}}=1}$ ${\displaystyle 0}$
30 ${\displaystyle {\tfrac {\pi }{6}}}$ ${\displaystyle {\tfrac {\sqrt {1}}{2}}={\tfrac {1}{2}}}$ ${\displaystyle {\tfrac {\sqrt {3}}{2}}}$ ${\displaystyle {\tfrac {1}{\sqrt {3}}}}$
45 ${\displaystyle {\tfrac {\pi }{4}}}$ ${\displaystyle {\tfrac {\sqrt {2}}{2}}={\tfrac {1}{\sqrt {2}}}}$ ${\displaystyle {\tfrac {\sqrt {2}}{2}}={\tfrac {1}{\sqrt {2}}}}$ ${\displaystyle 1}$
60 ${\displaystyle {\tfrac {\pi }{3}}}$ ${\displaystyle {\tfrac {\sqrt {3}}{2}}}$ ${\displaystyle {\tfrac {\sqrt {1}}{2}}={\tfrac {1}{2}}}$ ${\displaystyle {\sqrt {3}}}$
90 ${\displaystyle {\tfrac {\pi }{2}}}$ ${\displaystyle {\tfrac {\sqrt {4}}{2}}=1}$ ${\displaystyle {\tfrac {\sqrt {0}}{2}}=0}$ ${\displaystyle \infty }$
45–45–90
30–60–90

45–45–90度三角形、30–60–90度三角形以及正三角形是平面上的三種莫比斯三角形，任一內角都可以找到對應整數，使內角和整數的乘積為180，參照三角形群英语Triangle group

### 45–45–90度三角形

45–45–90度三角形為等腰直角三角形，在平面幾何中，這也是唯一是等腰三角形的直角三角形。不過在球面幾何學雙曲幾何中，有無限種也是等腰三角形的直角三角形。

### 30–60–90度三角形

30–60–90度三角形是平面幾何中唯一一個角度呈等差數列的直角三角形。其證明很簡單：假設三個角的角度為等差數列，可以表示為為α, α+δ, α+2δ，因為內角和為180°，可得3α+3δ = 180°，其中有一角會是60度，而且最大角需為90度，因此最小角會是30度。

### 角度呈等比數列的直角三角形

${\displaystyle \cos {\frac {\pi }{\varphi +1}}+\cos {\frac {\pi }{\varphi }}=0}$[註 2]

${\displaystyle e^{\frac {i\pi }{\varphi +1}}+e^{-{\frac {i\pi }{\varphi +1}}}+e^{\frac {i\pi }{\varphi }}+e^{-{\frac {i\pi }{\varphi }}}=0.\,}$

## 各邊有特殊關係

${\displaystyle m^{2}-n^{2}:2mn:m^{2}+n^{2}\,}$

### 常見的勾股数

3: 4 :5
5: 12 :13
8: 15 :17
7: 24 :25
9: 40 :41

3: 4 :5
5: 12 :13
8: 15 :17
7: 24 :25
9: 40 :41
11: 60 :61
12: 35 :37
13: 84 :85
15: 112 :113
16: 63 :65
17: 144 :145
19: 180 :181
20: 21 :29
20: 99 :101
21: 220 :221
24: 143 :145
28: 45 :53
28: 195 :197
32: 255 :257
33: 56 :65
36: 77 :85
39: 80 :89
44: 117 :125
48: 55 :73
51: 140 :149
52: 165 :173
57: 176 :185
60: 91 :109
60: 221 :229
65: 72 :97
84: 187 :205
85: 132 :157
88: 105 :137
95: 168 :193
96: 247 :265
104: 153 :185
105: 208 :233
115: 252 :277
119: 120 :169
120: 209 :241
133: 156 :205
140: 171 :221
160: 231 :281
161: 240 :289
204: 253 :325
207: 224 :305

### 斐波那契三角形

${\displaystyle 1:2:{\sqrt {5}}.}$

Andrew Clarke建議將長度比例為:${\displaystyle 1:2:{\sqrt {5}}}$的三角形稱為dom，因為此三角形可以由二格骨牌（domin）延對角線切割而成，此三角形是約翰·何頓·康威查爾斯·雷丁英语Charles Radin提出的非週期性英语aperiodic tiling風車貼磚英语pinwheel tiling的基礎。

### 幾乎等腰的直角三角形

a0 = 1, b0 = 2
an = 2bn–1 + an–1
bn = 2an + bn–1

an為斜邊的長度，n = 1, 2, 3, ....。最小的幾個三角形如下

3 : 4 : 5
20 : 21 : 29
119  : 120 : 169
696  : 697 : 985
4059  : 4060 : 5741
23660  : 23661 : 33461

## 註釋

1. ^ 根據黃金比例的定義，${\displaystyle {\frac {1}{\varphi }}=\varphi -1}$，由於${\displaystyle \sin {\frac {\pi }{2\varphi }}=\cos \left({\frac {\pi }{2}}-{\frac {\pi }{2\varphi }}\right)=\cos \left({\frac {\varphi \pi }{2\varphi }}-{\frac {\pi }{2\varphi }}\right)=\cos {\frac {(\varphi -1)\pi }{2\varphi }}=\cos {\frac {{\frac {1}{\varphi }}\pi }{2\varphi }}=\cos {\frac {\pi }{2\varphi ^{2}}}}$，因此原式成立。
2. ^ 根據黃金比例的定義，${\displaystyle 1+\varphi =\varphi ^{2}}$，因此${\displaystyle \cos {\frac {\pi }{\varphi +1}}=\cos {\frac {\pi }{\varphi ^{2}}}=-\cos \left(\pi -{\frac {\pi }{\varphi ^{2}}}\right)=-\cos \left({\frac {\varphi ^{2}\pi }{\varphi ^{2}}}-{\frac {\pi }{\varphi ^{2}}}\right)=-\cos {\frac {(\varphi ^{2}-1)\pi }{\varphi ^{2}}}=-\cos {\frac {\varphi \pi }{\varphi ^{2}}}=-\cos {\frac {\pi }{\varphi }}}$
或者，${\displaystyle {\frac {1}{\varphi +1}}+{\frac {1}{\varphi }}={\frac {1}{\varphi ^{2}}}+{\frac {1}{\varphi }}={\frac {1}{\varphi ^{2}}}+{\frac {\varphi }{\varphi ^{2}}}={\frac {1+\varphi }{\varphi ^{2}}}={\frac {1+\varphi }{1+\varphi }}=1}$，故${\displaystyle {\frac {\pi }{\varphi +1}}+{\frac {\pi }{\varphi }}=\pi }$，此兩角度互補，其cos值為相反數。

## 參考資料

1. ^ OEIS:A180014
2. ^ Weisstein, Eric W. (编). Rational Triangle. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. [2013-08-31]. （原始内容存档于2021-03-14） （英语）.
3. ^ A. Aleksei Petrovich Stakhov. Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer. World Scientific. 2009: p.86. ISBN 9812775838.
4. ^ C.C. Chen and T.A. Peng. Almost-isosceles right-angled triangles (PDF). University of Queensland. [2013-09-02]. （原始内容存档 (PDF)于2012-02-17）.