Definition:Pedal Triangle

From ProofWiki
Jump to navigation Jump to search

Definition

Pedal Triangle of Point with respect to Triangle

Let $\triangle ABC$ be a triangle.

Let $P$ be a point in the plane of $\triangle ABC$.

Let $PD$, $PE$ and $PF$ be perpendiculars dropped from $P$ to $BC$, $AC$ and $AB$ respectively.

Let $\triangle DEF$ be the triangle formed by the feet of the perpendiculars $PD$, $PE$ and $PF$.

Pedal-Triangle-of-Point.png $\qquad$ Pedal-Triangle-of-Point 2.png

$\triangle DEF$ is known as the pedal triangle of $P$ with respect to $\triangle ABC$.


Orthic (Pedal) Triangle

Let $\triangle ABC$ be a triangle.

Let $\triangle DEF$ be the triangle formed by the feet of the altitudes $AD$, $BC$ and $ED$ of $\triangle ABC$.

Orthic-Triangle.png

$\triangle DEF$ is known as the orthic triangle of $\triangle ABC$.

That is, the orthic triangle of $\triangle ABC$ is the pedal triangle of its orthocenter.


The orthic triangle of a given triangle $\triangle ABC$ is also known as the pedal triangle of $\triangle ABC$.

However, as this term is also used for the pedal triangle of any arbitrary point with respect to $\triangle ABC$, it is the policy of $\mathsf{Pr} \infty \mathsf{fWiki}$ to use the term orthic triangle consistently.


Also see

  • Results about pedal triangles can be found here.


Sources