Medians and Altitudes of Triangles

Median of a Triangle

A median is a segment from a vertex to the midpoint of the opposite side. Every Triangle has three medians.

Centroid

The three medians are concurrent at the centroid.

• The centroid divides each median in a 2:1 ratio from the vertex.
• The centroid is the center of gravity (balance Point) of the Triangle.
• The centroid is always inside the Triangle.
• Centroid formula: ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3)

Altitude of a Triangle

An altitude is a perpendicular segment from a vertex to the Line containing the opposite side (the base). Every Triangle has three altitudes.

Orthocenter

The three altitudes are concurrent at the orthocenter.

• In an acute Triangle, the orthocenter is inside.
• In a right Triangle, the orthocenter is at the right-Angle vertex.
• In an obtuse Triangle, the orthocenter is outside.

Summary of Triangle Centers