= Incircle of a triangle
{title2=$r=\Delta/s$}
= Incircle
{synonym}
The <incircle of a triangle> is the <circle> inside the <triangle> tangent to all three sides. Its center is the intersection of two internal angle bisectors; its radius is $r=\Delta/s$, where $\Delta$ is the area and $s$ the semiperimeter. To prove it is the largest contained <circle>, let any candidate have center $P$, radius $\rho$ and distances $d_i\geq\rho$ from the three side lines. Splitting the <triangle> into three <triangles> with vertex $P$ gives $\Delta=\frac12\sum_i\ell_i d_i\geq s\rho$. The tangent <circle> attains equality, proving maximality. For a <right triangle> with legs $a,b$ and hypotenuse $c$, $r=(a+b-c)/2$, since $r=ab/(a+b+c)$ and $a^2+b^2=c^2$.
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