三角形的五心

定义

正弦(sine)、馀弦(Cosine)定义

∠A之度数为α
sinα=对边长/斜边长
cosα=邻边长/斜边长

sinα=对边长/斜边长
cosα=邻边长/斜边长

1. sinα=cos(90°-α)
令∠B度数为β，β=90°-α，则sinα=a/c=cosβ=cos(90°-α)
2. cosα=sin(90°-α)
令∠B度数为β，β=90°-α，则cosα=b/c=sinβ=sin(90°-α)
3. sin2α+cos2α=1

${\displaystyle \sin ^{2}\alpha +\cos ^{2}\alpha ={\frac {a^{2}}{c^{2}}}+{\frac {b^{2}}{c^{2}}}={\frac {c^{2}}{c^{2}}}=1}$

两大公式

(一)圆周角等于对同弧圆心角的一半

正弦定理

${\displaystyle R}$为外接圆半径，则 ${\displaystyle {\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}=2R}$

(二)馀弦定理

${\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos \gamma }$
${\displaystyle b^{2}=c^{2}+a^{2}-2ca\cos \beta }$
${\displaystyle a^{2}=b^{2}+c^{2}-2bc\cos \alpha }$

三角形面积公式

(一)已知两边及其夹角

△=${\displaystyle {\frac {1}{2}}\times ab\times \sin \gamma }$

△=${\displaystyle {\frac {1}{2}}\times b\times h={\frac {1}{2}}\times b\times h\times {\frac {a}{a}}={\frac {1}{2}}\times a\times b\times {\frac {h}{a}}={\frac {1}{2}}\times ab\times \sin \gamma }$

(二)海伦公式

${\displaystyle s={\frac {a+b+c}{2}}}$

△=${\displaystyle {\sqrt {s\left(s-a\right)\left(s-b\right)\left(s-c\right)}}}$

(三)内切圆半径

${\displaystyle s={\frac {a+b+c}{2}}}$，r为内切圆半径

△面积 ${\displaystyle =sr}$

(四)外接圆半径

R为外接圆半径

△=${\displaystyle {\frac {abc}{4R}}}$

(六)三高

△= ½×a×ha= ½×b×hb= ½×c×hc

(七)以三点座标求面积

△=${\displaystyle \pm {\frac {1}{2}}{\begin{vmatrix}x_{2}-x_{1}&y_{2}-y_{1}\\x_{3}-x_{1}&y_{3}-y_{1}\end{vmatrix}}}$

△=${\displaystyle \pm {\frac {1}{2}}{\begin{vmatrix}x_{2}&y_{2}\\x_{3}&y_{3}\end{vmatrix}}}$

△=${\displaystyle {\frac {1}{2}}{\begin{vmatrix}x_{2}&y_{2}\\x_{3}&0\end{vmatrix}}=-{\frac {1}{2}}x_{3}y_{2}}$

△=${\displaystyle {\frac {1}{2}}|\mathbf {a} \times \mathbf {b} |}$