Area of Triangle in Terms of Circumradius

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

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, $$(ABC)=\frac{c\cdot h_c}{2}$$.

This gives us $$\frac{a\cdot b\cdot c}{4r}=(ABC)$$ as desired.