Coordinate System and Coordinates
13.0 Circumcentre of a triangle
13.0 Circumcentre of a triangle
The point of intersection of the perpendicular bisectors of the sides of a triangle is called the circumcenter.
As shown in figure, $ABC$ be the triangle with the vertices $A({x_1},{y_1})$, $B({x_2},{y_2})$ and $C({x_3},{y_3})$ whose sides $BC$, $CA$ and $AB$ are of lengths $a$, $b$ and $c$ respectively. $AD$, $BE$ and $CF$ are the perpendicular bisectors of sides $BC$, $CA$ and $AB$ respectively, then the co-ordinates of circumcentre of $\Delta ABC$ is $$\begin{equation} \begin{aligned} (\frac{{{x_1}\sin 2A + {x_2}\sin 2B + {x_3}\sin 2C}}{{sin2A + \sin 2B + \sin 2C}},\frac{{{y_1}\sin 2A + {y_2}\sin 2B + {y_3}\sin 2C}}{{sin2A + \sin 2B + \sin 2C}}) \\ {\text{or}} \\ (\frac{{a{x_1}\cos A + b{x_2}\cos B + c{x_3}\cos C}}{{a\cos A + b\cos B + c\cos C}},\frac{{a{y_1}\cos A + b{y_2}\cos B + c{y_3}\cos C}}{{a\cos A + b\cos B + c\cos C}}) \\\end{aligned} \end{equation} $$
Note:
- The circumcentre $O$ has equal distance with all the sides.
- Circumcentre of the obtuse triangle lies outside the triangle.
- Circumcentre of the right angled triangle lies on the midpoint of a hypotenuse of the triangle.
