Geometry

Some basic formulas involving triangles

$a^2 = b^2 + c^2 - 2bc \textrm{ cos } \alpha$
$b^2 = a^2 + c^2 - 2ac \textrm{ cos } \beta$
$a^2 = a^2 + b^2 - 2ab \textrm{ cos } \gamma$
$\frac{a}{\textrm{ sin } \alpha} = \frac{b}{\textrm{ sin } \beta} = \frac{c}{\textrm{ sin } \gamma} = 2R$

Median formulas

$m_a^2 = \frac{1}{4}( 2b^2 + 2c^2 - a^2 )$
$m_b^2 = \frac{1}{4}( 2a^2 + 2c^2 - b^2 )$
$m_c^2 = \frac{1}{4}( 2a^2 + 2b^2 - c^2 )$

Bisector formulas


$\frac{a}{b} = \frac{n}{m}$
$l^2 = ab - nm$

Right triangle formulas

$c^2 = a^2 + b^2$
$A = \frac{1}{2}a\cdot b = \frac{1}{2}c\cdot h$
$a^2 = n\cdot c$
$b^2 = m\cdot c$
$h^2 = n\cdot m$
$r = \frac{a + b - c}{2}$ - radius of inscribed circle
$\textrm{ sin }\alpha = \frac{a}{c}$
$\textrm{ cos }\alpha = \frac{b}{c}$
$\textrm{ tan }\alpha = \frac{a}{b}$
$\textrm{ cot }\alpha = \frac{b}{a}$

Area formulas

Let $p=\frac12(a+b+c)$

Area of triangle

$A = \frac{1}{2}ch_c$
$A = \frac{1}{2}ab \textrm{ sin } \gamma$
$A = \sqrt{p(p - a)(p - b)(p - c)}$
$A = pr$ with r we denote the radius of the triangle inscribed circle
$A = \frac{abc}{4R}$ - R is the radius of the prescribed circle

Area of parallelogram, rhombus

$A = AB\cdot DE = BC \cdot DF$
$A = AB \cdot AD \sin \alpha$
$A = \frac12 AC \cdot BD \sin \gamma$

Area of quadrilateral

$A = \frac12 AC \cdot BD \sin \varphi $

Area of prescribed polygon

$A = \frac12Pr$
P is the perimeter

Practice



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