Circumcircle of a Triangle - Practice Questions & MCQ

Circumcircle of a Triangle

The circumcircle of triangle ABC is the unique circle passing through the three vertices A, B, and C. Its centre, the circumcenter O, is the intersection of the perpendicular bisectors of the three sides. The circumradius is always denoted by R. 

The radius of the circumcircle of a ΔABC, R is given by the law of sines:

$\mathrm{R=\frac{\mathit{a}}{2\sin A}=\frac{\mathit{b}}{2\sin B}=\frac{\mathit{c}}{2\sin C}}$

The perpendicular bisector of the sides AB, BC and CA intersects at point O. So, O is the circumcentre and

$
O A=O B=O C=R
$
Let D be the midpoint of BC .

$
\begin{aligned}
& \angle \mathrm{BOC}=2 \angle \mathrm{BAC}=2 \mathrm{~A} \\
& \angle \mathrm{BOD}=\angle \mathrm{COD}=\mathrm{A}
\end{aligned}
$
So, in $\triangle O B D$,

$
\begin{aligned}
& \sin \mathrm{A}=\frac{\mathrm{BD}}{\mathrm{OB}}=\frac{a / 2}{\mathrm{R}}=\frac{a}{2 \mathrm{R}} \\
& \Rightarrow \mathrm{R}=\frac{a}{2 \sin \mathrm{~A}}
\end{aligned}
$

similarly,
$\mathrm{R}=\frac{b}{2 \sin \mathrm{~B}}$ and $\mathrm{R}=\frac{c}{2 \sin \mathrm{C}}$
$R$ can also be written in terms of the area of the triangle
Area of $\triangle A B C$,
$\Delta=\frac{1}{2} b \cdot \mathrm{c} \sin \mathrm{A}$
$\Rightarrow \sin \mathrm{A}=\frac{2 \Delta}{b c}$
and, $\mathrm{R}=\frac{a}{2 \sin \mathrm{~A}}$
From (i) and (ii)

$
\mathrm{R}=\frac{a b c}{4 \Delta}
$