Simson line
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In geometry, if one drops perpendiculars from a point P on the circumcircle of a triangle ABC to the sides (or their continuations) of the triangle, then the feet of the perpendiculars turn out to lie on a line, called the Simson line (of P for the triangle ABC).
The converse is also true; if the feet of the perpendiculars dropped from a point P to the sides of the triangle are collinear, then P is on the circumcircle. In other words, the Simson line of a point P is just the pedal triangle of P, in the case when that pedal triangle degenerates to a line.
The Simson line of a vertex of the triangle is the altitude of the triangle dropped from that vertex, and the Simson line of the point diametrically opposite to the vertex is the side of the triangle opposite to that vertex.
If P and P' are points on the circumcircle, then the angle between the Simson lines of P and P' is half the angle of the arc PP'. In particular, if the points are diametrically opposite, their Simson lines are perpendicular and in this case the intersection of the lines is on the nine-point circle.
Let H denote the orthocenter of the triangle ABC, then the Simson line of P bisects the segment PH in a point that lies on the nine-point circle.
Given two triangles with the same circumcircle, the angle between the Simson lines of a point P on the circumcircle for both triangles doesn't depend of P.
