Simson line

Builds on Triangle Circumcircle Collinear points Line

Definition

Let $ABC$ be a triangle in the Euclidean plane and let $\Gamma$ be its circumcircle. Let $P$ be a point of $\Gamma$ . Let $D$ , $E$ and $F$ be the feet of the perpendiculars from $P$ to the lines $BC$ , $CA$ and $AB$ . The points $D$ , $E$ and $F$ are collinear, and the Simson line of $P$ with respect to $ABC$ is the line that contains them.

Remarks

The collinearity in the definition is the Simson–Wallace theorem. Conversely, if the three feet from a point $P$ are collinear, then $P$ lies on $\Gamma$ .

Example: let $P = A'$ be the point of $\Gamma$ opposite to $A$ . Since $AA'$ is a diameter, the angles $\angle ABA'$ and $\angle ACA'$ are right angles, so the feet on $AB$ and $CA$ are $B$ and $C$ . The foot on $BC$ lies on the line $BC$ , so the Simson line of $A'$ is the line $BC$ .

Non-example: let $P$ be the circumcentre $O$ , which is not on $\Gamma$ . The feet of the perpendiculars from $O$ are the midpoints of the three sides. These form the medial triangle, so they are not collinear, and no Simson line is defined.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Circumcircle concept
Collinear points concept
Line concept
Triangle concept
Simson line definition

Tag	`D5VS0YFTEZ`
URI	`m2://D5VS0YFTEZ`
Kind	definition
Writing	W0 stub
MSC	51M04
Defines	Simson line
Uses	Triangle, Circumcircle, Collinear points, Line

Simson line

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here