 Incircle of a triangle

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 $d_i\ge \rho$ from the three side lines. Splitting the triangle into three triangles with vertex $P$ gives $\Delta =\frac{1}{2}{\sum }_i{\ell }_i{d}_i\ge s\rho$. 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 $a^2+b^2=c^2$.