# Triangle

Keywords: Triangle

Area of a triangle with height

$A=\frac{a{h}_{a}}{2}=\frac{b{h}_{b}}{2}=\frac{c{h}_{c}}{2}$
$A=\frac{ab\cdot sin\gamma }{2}=\frac{bc\cdot sin\alpha }{2}=\frac{ca\cdot sin\beta }{2}$

Area of a triangle - Heron's formula

$A=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)},\text{}\text{}\text{}s=\frac{a+b+c}{2}$

Area of a triangle using coordinates of points

$A=\frac{1}{2}\mid {x}_{1}\left({y}_{2}-{y}_{3}\right)+{x}_{2}\left({y}_{3}-{y}_{1}\right)+{x}_{3}\left({y}_{1}-{y}_{2}\right)\mid$

or the above formula using an absolute value of a determinant below

$A=\frac{1}{2}\left||\begin{array}{ccc}{x}_{1}& {y}_{1}& 1\\ {x}_{2}& {y}_{2}& 1\\ {x}_{3}& {y}_{3}& 1\end{array}|\right|$

INCIRCLE AND EXCIRCLES OF A TRIANGLE

Areao of triangel using radius of excircle

$A=\frac{a·b·c}{4R}$

Areao of triangel using radius of incircle

$A=s·r$