Area of a Triangle in Terms of Circumradius

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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.

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 $\triangle ABC$.

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \angle CAB$	$\cong$	$\displaystyle \angle CDB$	$\displaystyle $	$\displaystyle $	$\displaystyle $	by the Inscribed Angle Theorem
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle m\angle CBD$	$=$	$\displaystyle 90^\circ$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Angle Inscribed in Semicircle
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle m\angle AEC$	$=$	$\displaystyle 90^\circ$	$\displaystyle $	$\displaystyle $	$\displaystyle $	by the definition of altitude
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \angle CBD$	$\cong$	$\displaystyle \angle AEC$	$\displaystyle $	$\displaystyle $	$\displaystyle $

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

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \dfrac {h_c} b$	$=$	$\displaystyle \frac a {2r}$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle h_c$	$=$	$\displaystyle \frac {ab} {2r}$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \frac {c \cdot h_c} 2$	$=$	$\displaystyle \frac {a \cdot b \cdot c} {4r}$	$\displaystyle $	$\displaystyle $	$\displaystyle $

$(ABC) = \dfrac {c \cdot h_c} 2$

This gives us:

$\dfrac {a \cdot b \cdot c} {4r} = (ABC)$

as desired.

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \angle CAB\)	\(\cong\)	\(\displaystyle \angle CDB\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by the Inscribed Angle Theorem
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle m\angle CBD\)	\(=\)	\(\displaystyle 90^\circ\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Angle Inscribed in Semicircle
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle m\angle AEC\)	\(=\)	\(\displaystyle 90^\circ\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by the definition of altitude
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \angle CBD\)	\(\cong\)	\(\displaystyle \angle AEC\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \dfrac {h_c} b\)	\(=\)	\(\displaystyle \frac a {2r}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle h_c\)	\(=\)	\(\displaystyle \frac {ab} {2r}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \frac {c \cdot h_c} 2\)	\(=\)	\(\displaystyle \frac {a \cdot b \cdot c} {4r}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)