Medians, Altitudes, and Angle Bisectors

By the end of this lesson, you will be able to define and construct medians, altitudes, perpendicular bisectors, and angle bisectors of a triangle, and state key theorems about points on bisectors.

The Big Idea

Every triangle has four types of special segments: medians, altitudes, perpendicular bisectors of sides, and angle bisectors. Each set of three (one per side or angle) meets at a single point — the triangle centers. These centers have elegant properties and appear throughout mathematics and engineering.

Medians

A median connects a vertex to the midpoint of the opposite side.

Every triangle has exactly three medians. They always intersect at a single point — the centroid.

Properties:

• The centroid divides each median in the ratio 2:1 from vertex to midpoint
• The centroid is the triangle's center of mass (balance point)

Altitudes

An altitude is a perpendicular segment from a vertex to the line containing the opposite side.

Important: The altitude may fall outside the triangle for obtuse triangles (the "foot" falls on the extension of the opposite side).

All three altitudes meet at the orthocenter.

Perpendicular Bisectors of Sides

The perpendicular bisector of each side is perpendicular to that side at its midpoint. The three perpendicular bisectors meet at the circumcenter.

Perpendicular Bisector Theorem: Any point on the perpendicular bisector of a segment is equidistant from the endpoints.

Circumcenter property: The circumcenter is equidistant from all three vertices → it is the center of the circumscribed circle (circumcircle) that passes through all three vertices.

Angle Bisectors

The angle bisector from each vertex bisects that angle. The three angle bisectors meet at the incenter.

Angle Bisector Theorem: A point on the angle bisector of an angle is equidistant from the two sides of the angle.

Incenter property: The incenter is equidistant from all three sides → it is the center of the (incircle) that is tangent to all three sides.