Shortcut Trick
The point of intersection of perpendicular bisectors of the sides of a triangle is the circum-centre.
Memory Aid for Triangle Centers:
- Perpendicular Bisectors → Circum-centre (Center of circumcircle)
- Altitudes → Orthocentre
- Medians → Centroid
- Angle Bisectors → In-center (Center of incircle)
∴ The correct answer is circum-centre.
Alternate Method
Given: A triangle with its perpendicular bisectors intersecting at a common point.
Property Used: The circum-centre is the point that is equidistant from all three vertices of the triangle, allowing a circle (the circumcircle) to pass through them.
⇒ A perpendicular bisector is a line that divides a side into two equal parts and meets it at a 90° angle.
⇒ In any triangle, the three perpendicular bisectors are concurrent (they meet at a single point).
⇒ This point of concurrency is called the circum-centre.
⇒ The distance from the circum-centre to any vertex is the circum-radius (R).
∴ The correct answer is circum-centre.
Additional Information
Orthocentre
The point where the three altitudes (perpendiculars drawn from vertices to opposite sides) of a triangle intersect.
Centroid
The point where the three medians (lines connecting vertices to the midpoints of opposite sides) intersect; it divides each median in a 2:1 ratio.
In-center
The point of intersection of the internal angle bisectors; it is the center of the incircle which touches all sides of the triangle.
Euler Line
In any non-equilateral triangle, the orthocentre, centroid, and circum-centre are collinear, and the centroid divides the distance between the other two in a 2:1 ratio.