circumcircle

Problem
From the figure shown, ABC and DEF are equilateral triangles. Point E is the midpoint of AC and points D and F are on the circle circumscribing ABC. If AB is 12 cm, find DE.
 

The formula for the radius of the circle circumscribed about a triangle (circumcircle) is given by
 

where A_t is the area of the inscribed triangle.
 

This page will define the following: incenter, circumcenter, orthocenter, centroid, and Euler line.
 

Incenter
Incenter is the center of the inscribed circle (incircle) of the triangle, it is the point of intersection of the angle bisectors of the triangle.
 

The radius of incircle is given by the formula

where A_t = area of the triangle and s = ½ (a + b + c). See the derivation of formula for radius of incircle.