Point of concurrence:
Point of concurrence is the point where three or more lines meet.
A triangle has three sides, and each of the sides will have a perpendicular bisector.
For a better understanding, consider the figure given below.
Each of the sides of the triangle \(AB\), \(BC\), and \(AC\) has a perpendicular bisector in the figure.
At some point, the three perpendicular bisectors of the triangle meet.
The point at which the perpendicular bisectors meet is the point of congruence.
Here, the three perpendicular bisectors meet at \(G\). Hence, \(G\) becomes the point of concurrence.
The point of concurrence of the three perpendicular bisectors of the triangle is the circumcentre of the triangle. The circumcentre of the triangle is equidistant from the three vertices of the triangle.
How do we know if the point thus obtained is a circumcentre of a triangle or not?
1. The point obtained is a circumcentre if it is the point of concurrence of the three perpendicular bisectors of the triangle.
2. Similarly, with circumcentre as centre and, the distance between the circumcentre and any of the vertices as radius, draw a circle. If all three vertices of the triangle fall on the circle, then the point obtained is the circumcentre of the triangle.
3. The three perpendicular bisectors of a triangle are always concurrent.
4. The circumcentre of acute triangles lies inside the triangle. The circumcentre of the obtuse triangles lies outside the triangle. Similarly, the circumcentre of a right-angled triangle lies on the mid-point of the hypotenuse.