This page will define the following: incenter, circumcenter, orthocenter, centroid, and Euler line.
The radius of incircle is given by the formula
where At = area of the triangle and s = ½ (a + b + c). See the derivation of formula for radius of incircle.
As you can see in the figure above, circumcenter can be inside or outside the triangle. In the case of the right triangle, circumcenter is at the midpoint of the hypotenuse. Given the area of the triangle At, the radius of the circumscribing circle is given by the formula
You may want to take a look for the derivation of formula for radius of circumcircle.
The line that would pass through the orthocenter, circumcenter, and centroid of the triangle is called the Euler line.