GeoProof — Simson Line explorer

P on the circumcircle ⟹ the three feet are collinear

Drag P around the circle — the three feet always stay on one line (the Simson line). Drag a vertex to reshape the triangle.

The Simson Line

Take a triangle and its circumscribed circle. Pick a point P on that circle, and drop a perpendicular from P to each of the three sides (extending the sides as needed). The three feet of those perpendiculars are always collinear:

P on the circumcircle ⟹ the three feet lie on one line

That line is the Simson line of P. Move P off the circle and the three feet spread into a little triangle — the alignment is special to the circumcircle.

Special positions

When P is at a vertex, the Simson line becomes the altitude from that vertex. As P travels once around the circle, the Simson line sweeps around and turns through half as much — every direction in the plane is hit twice.

A hidden bisection

The Simson line of P always passes through the midpoint of the segment joining P to the triangle's orthocenter — linking this theorem to the nine-point circle.

Two Simson lines

For two points on the circle, the angle between their Simson lines is half the arc between the points. Diametrically opposite points give perpendicular Simson lines.

A bit of history

Although named after Robert Simson, the line was actually first published by William Wallace in 1799 — a common case of mis-attribution in mathematics.