Circumcentre and Orthocentre - Practice Questions & MCQ

Circumcentre

Perpendicular bisector of a side of a triabgle is the line through the midpoint of a side and perpendicular to it.  

The Circumcentre (O) of a triangle is the point of intersection of the perpendicular bisectors of the sides of a triangle.

Circumcentre is also defined as the center of a circle that passes through the vertices of a given triangle.

Coordinates of Circumcentre $(0)$ is

$
\left(\frac{\mathrm{x}_1 \sin 2 \mathrm{~A}+\mathrm{x}_2 \sin 2 \mathrm{~B}+\mathrm{x}_3 \sin 2 \mathrm{C}}{\sin 2 \mathrm{~A}+\sin 2 \mathrm{~B}+\sin 2 \mathrm{C}}, \frac{\mathrm{y}_1 \sin 2 \mathrm{~A}+\mathrm{y}_2 \sin 2 \mathrm{~B}+\mathrm{y}_3 \sin 2 \mathrm{C}}{\sin 2 \mathrm{~A}+\sin 2 \mathrm{~B}+\sin 2 \mathrm{C}}\right)
$

Orthocentre:

The Orthocentre (H) of a triangle is the point of intersection of altitudes which are drawn from one vertex to the opposite side of a triangle. 

Coordinates of Orthocentre (H) is

$
\left(\frac{\mathbf{x}_{\mathbf{1}} \tan \mathbf{A}+\mathbf{x}_2 \tan \mathbf{B}+\mathbf{x}_{\mathbf{3}} \tan \mathbf{C}}{\tan \mathbf{A}+\tan \mathbf{B}+\tan \mathbf{C}}, \frac{\mathbf{y}_{\mathbf{1}} \tan \mathbf{A}+\mathbf{y}_{\mathbf{2}} \tan \mathbf{B}+\mathbf{y}_{\mathbf{3}} \tan \mathbf{C}}{\tan \mathbf{A}+\tan \mathbf{B}+\tan \mathbf{C}}\right)
$
Note:
For a right angled triangle, the orthocenter is the vertex containing the right angle