Simson Line Theorem

In Euclidean geometry, the Simson line theorem states that the three closest points from a point $P$ on the circumcircle on $\triangle ABC$ to sides (or the extensions of) $AB$, $AC,$ and $BC$ are collinear.

The converse is also true: if the $3$ closest points from the three sides of a triangle to a point $P$ is collinear, then $P$ lies on the circumcircle of the triangle.

Let $P$ be a point on the circumcircle of $\triangle ABC.$ Points $D, E,$ and $F$ are the feet of the perpendicular bisectors from $P$ to sides (or the extensions of) $AB$, $AC,$ and $BC,$ respectively. Then $D, E,$ and $F$ are collinear.

Simson Line

The trilinear equation of the Simson line for a point $p:q:r$ lying on the circumcircle satisfies $cpq+bpr+aqr=0$.
The Simson line bisects the line $HP$, where $H$ is the orthocenter.
Moreover, the midpoint of $HP$ lies on the nine-point circle.
The Simson lines of two opposite point on the circumcircle of a triangle are perpendicular and meet on the nine-point circle.

• Pedal Triangle