Altitudes

Definition: Altitude

An altitude in a triangle is a line segment $h$ with the following properties:

• The endpoints of $h$ are a vertex $V$ of the triangle and a point on the extension of the side $s$ opposite to $V$.

• The segment $h$ is perpendicular to the extension of $s$.

Definition: Base

The side which an altitude is perpendicular to is known as the altitude’s base.

NOTATION

If the base of an altitude is $s$, we often denote the altitude itself with $h_s$.

Theorem: Orthocenter

The extensions of the three altitudes in a given triangle are concurrent.

PROOF

TODO

Definition: Orthocenter

The point of intersection of the extensions of a triangle’s altitudes is known as the triangle’s orthocenter.