In an equilateral triangle, the circumcenter coincides with which of the following?

Correct — C. In an equilateral triangle every median is also a perpendicular bisector.

A median joins a vertex to the midpoint of the opposite side
The two halves it makes are congruent (SSS), so it meets that side at 90°
So the three medians and the three perpendicular bisectors are the same three lines

Centroid = where the medians meet
Circumcentre = where the perpendicular bisectors meet
Same lines, same point → option (c)

Why the others are wrong

• (a)
  A point outside the triangle — The circumcentre falls outside the triangle when one angle is obtuse. Every angle of an equilateral triangle is 60°, so its circumcentre is inside.
• (b)
  The midpoint of one of its sides — The circumcentre sits at the midpoint of a side in a right triangle, where that side is the hypotenuse. An equilateral triangle has no right angle.
• (d)
  One of the vertices — The circumcentre is equidistant from all three vertices, so it can never be a vertex. The centre that sits on a vertex, in a right triangle, is the orthocentre.

A triangle has four classic centres. The centroid is where the medians meet, the circumcentre where the perpendicular bisectors meet, the incentre where the angle bisectors meet, and the orthocentre where the altitudes meet.

In an equilateral triangle each median is also an altitude, an angle bisector and a perpendicular bisector. So all four centres coincide, at the point that divides each median 2 : 1 from the vertex.

Key facts

• In an equilateral triangle the centroid, circumcentre, incentre and orthocentre are one point.
• The circumcentre lies inside an acute triangle, at the midpoint of the hypotenuse of a right triangle, and outside an obtuse triangle.
• For an equilateral triangle of side a, the circumradius is a⁄√3 and the inradius is a⁄(2√3).
• The centroid divides each median in the ratio 2 : 1, measured from the vertex.

Common traps

• Confusing the orthocentre with the circumcentre. In a right triangle the orthocentre is the right-angle vertex and the circumcentre is the midpoint of the hypotenuse.
• Assuming the centres coincide in any isosceles triangle. There they lie on the axis of symmetry but are separate points.

No directly related past PYQ was found.