Triangle Perpendicular Bisector and Circumcenter
Understand perpendicular bisectors, the circumcenter, and the circumscribed circle
Overview
A perpendicular bisector of a triangle is a line that passes through the midpoint of a side and is perpendicular to that side. This is a fundamental concept in triangle geometry with important applications.
The perpendicular bisector of a side of a triangle is the line that:
- Passes through the midpoint of the side
- Forms a 90° angle with that side
- Is equidistant from the two endpoints of the side
Every triangle has exactly three perpendicular bisectors, one for each side. All three perpendicular bisectors intersect at a single point called the circumcenter.
The Circumcenter
The circumcenter is the point where all three perpendicular bisectors of a triangle intersect. It has special geometric significance.
Circumcenter Properties
- The circumcenter is the intersection of all three perpendicular bisectors
- Every triangle has exactly one circumcenter
- The circumcenter is equidistant from all three vertices
- This distance equals the radius of the circumscribed circle (circumcircle)
- The circumcenter can be inside, on, or outside the triangle
- For an acute triangle, the circumcenter is inside
- For a right triangle, the circumcenter is on the hypotenuse (at its midpoint)
- For an obtuse triangle, the circumcenter is outside
Visual Example
Circumcenter Position by Triangle Type
The location of the circumcenter depends on the type of triangle. This is similar to the orthocenter, but with different rules.
Acute Triangle
All angles < 90°
Circumcenter location: Inside the triangle
The circumcenter lies within the triangle's interior. The circumcircle passes through all three vertices and contains the entire triangle.
Right Triangle
One angle = 90°
Circumcenter location: On the hypotenuse (at its midpoint)
The circumcenter is located exactly at the midpoint of the hypotenuse. The hypotenuse is a diameter of the circumcircle, and the right angle vertex lies on the circumcircle.
Obtuse Triangle
One angle > 90°
Circumcenter location: Outside the triangle
The circumcenter lies outside the triangle, on the opposite side of the longest side from the obtuse angle. The circumcircle still passes through all three vertices.
The Circumscribed Circle (Circumcircle)
The circumcenter is the center of the circumscribed circle, also called the circumcircle. This circle passes through all three vertices of the triangle.
- The circumcircle passes through all three vertices of the triangle
- The center is the circumcenter (intersection of perpendicular bisectors)
- The radius (called the circumradius) equals the distance from the circumcenter to any vertex
- Every triangle has a unique circumcircle
- The triangle is inscribed in the circumcircle
Key Properties
Perpendicular bisectors and the circumcenter have several important geometric properties.
Fundamental Properties
- Each perpendicular bisector is equidistant from the endpoints of its side
- The perpendicular bisectors always meet at one point (the circumcenter)
- Any point on a perpendicular bisector is equidistant from the endpoints of that side
- The circumcenter is equidistant from all three vertices
- This common distance is the circumradius of the triangle
- The circumcenter location depends on the triangle's angle types
Comparison: Triangle's Special Points
Circumcenter
From: Perpendicular bisectors
Property: Equidistant from vertices
Circle: Circumcircle (around triangle)
Incenter
From: Angle bisectors
Property: Equidistant from sides
Circle: Incircle (inside triangle)
Centroid
From: Medians
Property: Center of mass (2:1 division)
Position: Always inside
Orthocenter
From: Altitudes
Property: Perpendicular meeting point
Position: Inside/on/outside
Key Points
- A perpendicular bisector passes through the midpoint of a side and is perpendicular to it
- Every triangle has exactly three perpendicular bisectors
- All three perpendicular bisectors meet at the circumcenter
- The circumcenter is equidistant from all three vertices
- In an acute triangle, the circumcenter is inside the triangle
- In a right triangle, the circumcenter is on the hypotenuse's midpoint
- In an obtuse triangle, the circumcenter is outside the triangle
- The circumcenter is the center of the circumscribed circle (circumcircle)
- The circumcircle passes through all three vertices of the triangle
Online Calculator
Use the interactive calculator to compute perpendicular bisector properties and circumcircle measurements:
Triangle Bisector Calculator →
|
|