For a given triangle and a point P on its circumcircle, the orthogonal projections (feet) from P to the three sides of the triangle (or their extensions) lie on a single straight line; that common line is called the Simson line of P with respect to the triangle.

Principle

The pedal points of a circumferential point to the sides of a triangle are collinear: perpendicular projection from a point on the circumcircle produces three aligned feet.

Demonstration

Take triangle ABC and let P be a point on its circumcircle. Drop perpendiculars from P to lines AB, BC and CA; denote the feet by D, E and F. The Simson line assertion states that D, E and F are collinear. Concretely, one may construct ABC, draw its circumcircle, choose P on that circle, and verify by coordinate or synthetic geometry that the three perpendicular feet lie on one line.

Misapplication

Assuming the same collinearity when P is off the circumcircle; for a generic external or internal point not on the circumcircle the three perpendicular feet need not be collinear and typically are not.

Consequence

When applicable, the Simson line provides a linear locus associated to P that is useful for constructions and for proving further collinearities and concurrency results in triangle geometry; its envelope and intersections with triangle centers encode classical correspondences.

Reversal

The converse holds: if the three perpendicular feet from a point P to the (extended) sides of triangle ABC are collinear, then P lies on the circumcircle of ABC; thus the property characterizes circumcircular points.

Boundary

Applies only to nondegenerate plane triangles and to points P lying on the triangle's circumcircle; projections to the extended sides are allowed, but degenerate triangles or projections at vertices require separate handling.

Semantic Tension

Often confused with other named lines in triangle geometry (for example certain pedal or Steiner lines); the Simson line is distinguished by the requirement that P lies on the circumcircle rather than by other alignment or midpoint properties.

Synthesis

The Simson line is the straight-line locus formed by the three perpendicular feet from any point on a triangle's circumcircle; it both characterizes circumcircular points by collinearity of pedal points and furnishes a compact tool for many classical triangle constructions.