Triangle centres: The circumcentre of a triangle is the point of intersection of which lines?

Introduction / Context:
Every triangle has several famous “centres”: centroid, incenter, circumcenter, and orthocenter. Each is defined by the concurrency (intersection) of a special family of lines. Correctly matching the family to the associated centre is a core geometry skill.

Given Data / Assumptions:

• We are asked about the circumcentre.
• We repair ambiguous wording by listing standard line families explicitly.

Concept / Approach:
The circumcentre is the centre of the unique circle passing through all three vertices (circumcircle). Points equidistant from the endpoints of a side lie on that side’s perpendicular bisector. A point equidistant from all three vertices must lie at the common intersection of the three perpendicular bisectors.

Step-by-Step Solution:
Construct the perpendicular bisector of side AB.Construct the perpendicular bisector of side BC; their intersection is equidistant from A, B, C.The third bisector (of CA) passes through the same point by concurrency, defining the circumcentre.

Verification / Alternative check:
From the intersection O, OA = OB = OC; O is the circle centre with radius OA.

Why Other Options Are Wrong:
Medians meet at the centroid; angle bisectors at the incenter; altitudes at the orthocenter.

Common Pitfalls:
Confusing “perpendicular bisectors” with “angle bisectors”. They are different; the former are perpendicular to sides, the latter split interior angles.

Final Answer:
Perpendicular bisectors