Area of Triangle in Terms of Circumradius

Theorem
The area of $\triangle ABC$ is given by the formula:
 * $(ABC) = \dfrac {a \cdot b \cdot c} {4r}$

where $r$ is the circumradius and $a, b, c$ are the sides.

Proof

 * [[File:Geo1231.PNG]]

Let $O$ be the circumcenter of $\triangle ABC$.

Let $E$ be the foot of the altitude from $C$.

Construct a point $D$ at the opposite endpoint of the diameter from $C$ on the circumcircle of $\triangle ABC$.

Then by AA similarity $\triangle AEC \sim \triangle DBC$

By Area of a Triangle in Terms of Side and Altitude:
 * $(ABC) = \dfrac {c \cdot h_c} 2$

This gives us:
 * $\dfrac {a \cdot b \cdot c} {4r} = (ABC)$

as desired.