Definition
The circumcentre is the point equidistant from the vertices
Geometry • Triangle • Triangle Centres
Statement
The circumcentre of a triangle is the common point of the perpendicular bisectors of its sides, and it is the unique point of the plane equidistant from all three vertices, the common distance being the circumradius . The incentre, by contrast, is the unique point equidistant from the three SIDES; the centroid is the common point of the medians and the orthocentre the common point of the altitudes, and neither of those two is equidistant from the vertices except in an equilateral triangle, where all four centres coincide.
How to use it
- 1Whenever a condition forces one point to be the same distance from all three vertices, that point is the circumcentre - there is no other.
- 2Equidistant from the SIDES means the incentre; equidistant from the VERTICES means the circumcentre. This is the pair an examiner tests.
- 3In a right triangle the circumcentre is the midpoint of the hypotenuse, so is half the hypotenuse; in an obtuse triangle it falls outside the triangle.
Worked example
A point satisfies for a triangle . What is the circumradius of ?
Step-by-step
- 1
Why: Only the circumcentre is equidistant from all three vertices.
- 2
Why: The common distance from the circumcentre to the vertices is the circumradius.
Answer
Watch out
The circumcentre need not lie inside the triangle: it is inside for an acute triangle, on the hypotenuse for a right triangle and outside for an obtuse one. Confusing it with the incentre, which is equidistant from the sides and always inside, is the standard trap.