The three perpendicular bisectors of a triangle meet at circumcenter, the center of the circle that passes through the three vertices.

Who discovered

The construction is the proposition IV.5 of The Elements of Euclid (c. 300 BC): “circumscribe a circle to a given triangle”. The meeting point of perpendicular bisectors is at the same distance from the three vertices, and therefore serves as the center. Euclid also observes where the center is: inside the triangle if it is acute, on the largest side if it is rectangular and outside if it is obtuse.

Euclid, Renaissance representation traditionally associated with Euclid or Archimedes.
Euclid, Renaissance representation traditionally associated with Euclid or Archimedes.Rafael Sanzio · Wikimedia Commons · Public domain

The story

The same problem appears in another form: through three points that are not aligned, exactly one circle passes. This is the basis of many localization and reconstruction techniques.

The meeting of perpendicular bisectors is at the same distance from the three vertices and determines the circumscribed circle.
The meeting of perpendicular bisectors is at the same distance from the three vertices and determines the circumscribed circle.Original didactic illustration from the blog.

Archaeologists use circumcenter to reconstruct broken pottery pieces. With a single fragment of the edge of a plate, they mark three points, trace two perpendicular bisectors and find the center and radius of the original plate — and thus know its size.

In everyday life

  • Reconstruct a plate or a wheel. Three points on the edge are enough to find the center and diameter.
  • Equidistant place. A hospital, an antenna or a meeting point at the same distance from three cities is at circumcenter.
  • Draw an arc through three points. Drawing and CAD programs use this construction in the “arc through three points” tool.
  • Right triangle. In a right triangle, circumcenter is the midpoint of the hypotenuse: the corner of a square always “sees” the hypotenuse as the diameter.

Sources: Elements, proposition IV.5 · Euclid (MacTutor).

Theory and constructions

TeoremaCompetition from perpendicular bisectors

The perpendicular bisectors of the sides of a triangle intersect at the same point, which is equal distance from the vertices:

$$m_1 \cap m_2 \cap m_3 = \lbrace O \rbrace, \qquad OA = OB = OC.$$

Demonstration. Let $O$ be the meeting of $m_2$ and $m_3$ (perpendicular bisectors of $\overline{AC}$ and $\overline{AB}$). Like $O \in m_2$, $OA = OC$; such as $O \in m_3$, $OA = OB$. So $OB = OC$, and $O$ is in perpendicular bisector $m_1$ of $\overline{BC}$.

Definiçãocircumcenter

The meeting point of the perpendicular bisectors sides is circumcenter. It is the center of the circle circumscribed to the triangle (which passes through the three vertices).