Derivation of Formula for Radius of Incircle

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The radius of incircle is given by the formula
 

$r = \dfrac{A_t}{s}$

 

where At = area of the triangle and s = semi-perimeter.
 

Derivation
 

Figure for derivation of radius of incircle

 

Let
At = Area of triangle ABC
At = Area of triangle BOC + Area of triangle AOC + Area of triangle AOB
 

$A_t = A_{BOC} + A_{AOC} + A_{AOB}$

$A_t = \frac{1}{2}ar + \frac{1}{2}br + \frac{1}{2}cr$

$A_t = \frac{1}{2}(a + b + c)\,r$
 

Let   $\frac{1}{2}(a + b + c) = s$,   the semi-perimeter
 

Thus,
$A_t = sr$

$r = \dfrac{A_t}{s}$

 

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