Altitudes

Definition: Altitude

An altitude in a [[Triangles|triangle]] is a [[Line Segments|line segment]] \(h\) with the following properties:

  • The endpoints of \(h\) are a [[Polygons|vertex]] \(V\) of the triangle and a [[Euclidean Geometry|point]] on the [[Line Segments|extension]] of the [[Polygons|side]] \(s\) opposite to \(V\).
  • The segment \(h\) is [[Angle between Lines|perpendicular]] to the extension of \(s\).

Definition: Base

The [[Polygons|side]] which an [[Altitudes|altitude]] is [[Angle between Lines|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 [[Line Segments|extensions]] of the three [[Altitudes]] in a given [[Triangles|triangle]] are [[Concurrent Lines|concurrent]].

Proof

TODO

Definition: Orthocenter

The [[Euclidean Geometry|point]] of intersection of the [[Line Segments|extensions]] of a [[Triangles|triangle]]'s [[Altitudes]] is known as the triangle's orthocenter.