##  [Simson Line](/simson-line-0) 

 Definition

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

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

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

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

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

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

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

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

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.