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How do perpendicular bisectors locate a triangle's circumcenter?

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Map of the Greco-Roman world highlighting the birthplaces of notable ancient mathematicians, marked with the pi symbol and labeled with their names. Cities such as Athens, Alexandria, Syracuse, and Miletus are shown around the Mediterranean, with modern country boundaries outlined. An inset provides detail for the region of Greece and western Anatolia. — geometry This map spans a millennium of prominent Greco-Roman mathematicians, from Thales of Miletus (c. 600 bce) to Hypatia of Alexandria (c. 400 ce).

In geometry, a perpendicular bisector is defined as a line that extends from the midpoint of a triangle's side at a right angle.1

The circumcenter is located at the unique point where the three perpendicular bisectors of a triangle's sides intersect. This point is significant because it is equidistant from all three vertices of the triangle, serving as the center of the circumcircle—the circle that passes through each of those vertices.21