Circumcenter — Perpendicular Bisector Condition

Circumcenter O is the intersection of perpendicular bisectors of the sides.

The circumcenter $O$ is equidistant from all three vertices: $OA = OB = OC = R$ (circumradius). It is the intersection of the perpendicular bisectors of the sides.

Construction

Perpendicular bisector of $AB$: passes through the midpoint $M_{AB} = \left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2}\right)$ and is perpendicular to $AB$.

Slope of $AB$: $m_{AB} = \dfrac{y_2-y_1}{x_2-x_1}$.

Equation of perpendicular bisector of $AB$:

$$y - \frac{y_1+y_2}{2} = -\frac{1}{m_{AB}}\left(x - \frac{x_1+x_2}{2}\right)$$

Write the perpendicular bisector of $BC$ similarly. Their intersection is $O$.

Alternative via distance condition

$OA^2 = OB^2$ gives one equation, $OB^2 = OC^2$ gives another. Solving these two simultaneously also yields $O$ — and avoids computing slopes when coordinates are messy.

Key facts

• All three perpendicular bisectors are concurrent at $O$.
• For an acute triangle, $O$ is inside; obtuse triangle, $O$ is outside; right triangle, $O$ is the midpoint of the hypotenuse.
• $R = \dfrac{abc}{4\Delta}$ where $a,b,c$ are side lengths and $\Delta$ is the area.