Circumcenter of a Triangle Calculator

What Is the Circumcenter of a Triangle?

The circumcenter of a triangle is the point where the three perpendicular bisectors of the triangle's sides intersect. It is designated by the letter $U$ (or $O$) and serves as the exact center of the circumscribed circle (or circumcircle), which is the unique circle that passes through all three vertices of the triangle.

Because the circumcenter is equidistant from all three vertices $A$, $B$, and $C$, the distance from the circumcenter to any vertex is equal to the triangle's circumradius ($R$):

$$R = |UA| = |UB| = |UC|$$

Circumcenter Formulas and Coordinate Geometry

Given three vertices in the Cartesian plane $A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$, you can calculate the exact circumcenter coordinates $(U_x, U_y)$ using the system of perpendicular bisectors or Cramer's determinant method:

First, compute the denominator determinant $D$:

$$D = 2 \cdot [x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)]$$

If $D = 0$, the three points are collinear and cannot form a triangle. Otherwise, the coordinates $U_x$ and $U_y$ are:

$$U_x = \frac{(x_1^2 + y_1^2)(y_2 - y_3) + (x_2^2 + y_2^2)(y_3 - y_1) + (x_3^2 + y_3^2)(y_1 - y_2)}{D}$$

$$U_y = \frac{(x_1^2 + y_1^2)(x_3 - x_2) + (x_2^2 + y_2^2)(x_1 - x_3) + (x_3^2 + y_3^2)(x_2 - x_1)}{D}$$

Circumradius Formula from Side Lengths and Area

When you know the three side lengths $a$, $b$, and $c$, the circumradius $R$ can be found directly without coordinates using the triangle's area:

$$R = \frac{a \cdot b \cdot c}{4 \cdot \text{Area}}$$

Where the area is calculated using Heron's formula:

$$s = \frac{a + b + c}{2}$$

$$\text{Area} = \sqrt{s(s - a)(s - b)(s - c)}$$

Position of the Circumcenter Relative to Triangle Type

The location of the circumcenter depends entirely on the angle classification of the triangle:

• Acute Triangle: All three angles are less than 90 degrees. The circumcenter lies strictly inside the triangle.
• Right Triangle: One angle is exactly 90 degrees. The circumcenter lies directly on the midpoint of the hypotenuse, and the circumradius is exactly half the hypotenuse length ($R = c / 2$).
• Obtuse Triangle: One angle is greater than 90 degrees. The circumcenter lies outside the triangle opposite the obtuse angle.

Circumcircle Equation

Once you have the circumcenter coordinates $(U_x, U_y)$ and the circumradius $R$, you can express the equation of the circumscribed circle in Standard Form:

$$(x - U_x)^2 + (y - U_y)^2 = R^2$$

Expanding this equation yields the General Form:

$$x^2 + y^2 - 2U_x x - 2U_y y + (U_x^2 + U_y^2 - R^2) = 0$$

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