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Circumcenter

Circumcenter

The center of a circumcircle. For any circumscribable polygon, the circumcenter is found at the point of intersection of the perpendicular bisectors of the sides.

 

A triangle with a circumcircle; the circumcenter is marked at the intersection of perpendicular bisectors, with arrows...

 

 

See also

Centers of a triangle

Key Formula

Circumcenter O=intersection of perpendicular bisectors of the sides\text{Circumcenter } O = \text{intersection of perpendicular bisectors of the sides} For a triangle with vertices A(x1,y1)A(x_1, y_1), B(x2,y2)B(x_2, y_2), C(x3,y3)C(x_3, y_3), the circumcenter (h,k)(h, k) satisfies: (h−x1)2+(k−y1)2=(h−x2)2+(k−y2)2=(h−x3)2+(k−y3)2(h - x_1)^2 + (k - y_1)^2 = (h - x_2)^2 + (k - y_2)^2 = (h - x_3)^2 + (k - y_3)^2
Where:
  • OO = The circumcenter point
  • (h,k)(h, k) = Coordinates of the circumcenter
  • (x1,y1),(x2,y2),(x3,y3)(x_1, y_1), (x_2, y_2), (x_3, y_3) = Coordinates of the triangle's three vertices
  • RR = Circumradius — the distance from the circumcenter to any vertex

Worked Example

Problem: Find the circumcenter of the triangle with vertices A(0, 0), B(6, 0), and C(0, 8).
Step 1: Set up the equal-distance equations. The circumcenter (h, k) is equidistant from all three vertices. Start by setting the squared distance from A equal to the squared distance from B.
(h−0)2+(k−0)2=(h−6)2+(k−0)2(h - 0)^2 + (k - 0)^2 = (h - 6)^2 + (k - 0)^2
Step 2: Expand and simplify the first equation to solve for h.
h2+k2=h2−12h+36+k2  ⟹  12h=36  ⟹  h=3h^2 + k^2 = h^2 - 12h + 36 + k^2 \implies 12h = 36 \implies h = 3
Step 3: Now set the squared distance from A equal to the squared distance from C.
h2+k2=h2+(k−8)2=h2+k2−16k+64h^2 + k^2 = h^2 + (k - 8)^2 = h^2 + k^2 - 16k + 64
Step 4: Simplify to solve for k.
16k=64  ⟹  k=416k = 64 \implies k = 4
Step 5: The circumcenter is at (3, 4). Verify by computing the distance to each vertex: the circumradius R should be the same for all three.
R=32+42=5,(3−6)2+42=5,32+(4−8)2=5  ✓R = \sqrt{3^2 + 4^2} = 5, \quad \sqrt{(3-6)^2 + 4^2} = 5, \quad \sqrt{3^2 + (4-8)^2} = 5 \; \checkmark
Answer: The circumcenter is at (3, 4) with circumradius R = 5.

Another Example

This example uses the perpendicular-bisector-intersection method (finding equations of perpendicular bisectors and solving simultaneously), rather than the algebraic equal-distance approach used in the first example. It demonstrates an alternative geometric technique.

Problem: Find the circumcenter of the triangle with vertices A(1, 1), B(5, 1), and C(1, 7).
Step 1: Find the midpoint and slope of side AB. The midpoint is (3, 1) and the slope of AB is 0 (horizontal line). The perpendicular bisector is therefore a vertical line.
Perpendicular bisector of AB:  x=3\text{Perpendicular bisector of } AB: \; x = 3
Step 2: Find the midpoint and slope of side AC. The midpoint is (1, 4) and the slope of AC is undefined (vertical line). The perpendicular bisector is therefore a horizontal line.
Perpendicular bisector of AC:  y=4\text{Perpendicular bisector of } AC: \; y = 4
Step 3: The circumcenter is the intersection of these two perpendicular bisectors.
(h,k)=(3,4)(h, k) = (3, 4)
Step 4: Compute the circumradius by finding the distance from (3, 4) to vertex A(1, 1).
R=(3−1)2+(4−1)2=4+9=13R = \sqrt{(3-1)^2 + (4-1)^2} = \sqrt{4 + 9} = \sqrt{13}
Answer: The circumcenter is at (3, 4) with circumradius R = √13.

Frequently Asked Questions

Can the circumcenter be outside the triangle?
Yes. For an obtuse triangle, the circumcenter lies outside the triangle, on the opposite side of the longest edge from the obtuse angle. For a right triangle, it falls exactly on the hypotenuse's midpoint. Only for an acute triangle does the circumcenter sit inside the triangle.
What is the difference between the circumcenter and the incenter?
The circumcenter is equidistant from all vertices and is found using perpendicular bisectors of the sides. The incenter is equidistant from all sides and is found using angle bisectors. The circumcircle passes through the vertices, while the incircle is tangent to the sides. These are generally two different points unless the triangle is equilateral, in which case they coincide.
How do you find the circumradius from the circumcenter?
Once you know the circumcenter coordinates (h, k), compute its distance to any one vertex using the distance formula: R=(h−x1)2+(k−y1)2R = \sqrt{(h - x_1)^2 + (k - y_1)^2}. Alternatively, for a triangle with sides a, b, c and area K, the circumradius is R=abc4KR = \frac{abc}{4K}.

Circumcenter vs. Incenter

CircumcenterIncenter
DefinitionCenter of the circumscribed circle passing through all verticesCenter of the inscribed circle tangent to all sides
Found usingPerpendicular bisectors of the sidesAngle bisectors of the interior angles
Equidistant fromAll vertices of the polygonAll sides of the polygon
Location in a triangleInside (acute), on hypotenuse (right), outside (obtuse)Always inside the triangle
Associated radiusCircumradius RInradius r

Why It Matters

The circumcenter appears throughout geometry courses whenever you need to find a circle through given points — for example, determining the equation of a circle passing through three non-collinear points. It is also central to triangle geometry proofs, the Euler line (which connects the circumcenter, centroid, and orthocenter), and real-world applications like finding the optimal location equidistant from three landmarks.

Common Mistakes

Mistake: Confusing perpendicular bisectors with angle bisectors when constructing the circumcenter.
Correction: Perpendicular bisectors are lines that cross each side at its midpoint at 90°. Angle bisectors split interior angles in half and lead to the incenter, not the circumcenter. Always bisect the sides perpendicularly.
Mistake: Assuming the circumcenter is always inside the triangle.
Correction: The circumcenter lies inside only for acute triangles. For an obtuse triangle it is outside, and for a right triangle it is exactly at the midpoint of the hypotenuse. Always check the triangle type before drawing conclusions about the circumcenter's position.

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