Simson Line

Perpendicular feet from a point are collinear exactly on the circumcircle.

The idea

Theorem (Simson line).

Let $P$ be a point and $ABC$ a triangle, and let $X$, $Y$, $Z$ be the feet of the perpendiculars from $P$ to the lines $BC$, $CA$, $AB$. Then $X$, $Y$, $Z$ are collinear if and only if $P$ lies on the circumcircle of $ABC$.

The circumcircle is the circle through the three vertices. A foot may fall beyond the end of a side, which is why the perpendiculars are dropped to the side-lines rather than to the segments. For most positions of $P$ the three feet form a triangle of their own, the pedal triangle of $P$; the theorem identifies exactly the points for which that triangle degenerates to a line. The line through the three feet is then the Simson line of $P$.

Ways to work on it

• Walkthrough. Why concyclic feet line up: the circumcircle criterion.
• Practice. Decide collinearity from where the point sits.
• Hardest. Apply the converse and the cyclic-quadrilateral structure.

Not sure where to start? Take the ten-question placement test.