Simson's Line Thm

  Given any triangle ABC, draw the circumcircle and label any point on the circle P. The feet of the perpendiculars from P to the three sides of the triangle will all lie on a line. This line is called the Simson Line
Here are several properties of the Simson line. From point P, construct a segment to the orthocenter. The midpont of P-Orthocenter is on the Simson line. The point is also on the Nine point circle. The Simson lines for two points on the opposite ends of a diameter are perpendicular to each other, and meet on the Nine Point Circle.

The theorem is named for Scottish Mathematician Robert Simson (1687-1768). At the suggestion of Edmond Halley, for whom the comet is named, Simson devoted himself to the restoration of the work of the early Greek geometers, such as Euclid and Apollonius of Perga. Although Simson did make numerous discoveries in geometry and may have found a proof of this one, he certainly was not the first to do so. There seems to be no record of him publishing the theorem, and his name is attached because it was (mis?)attributed to Simson by Servois in Gergonne's Journal.
The rightful credit probably belongs to William Wallace (1768-1843), also Scottish, and a self taught mathematician. He is credited with publishing the theorem in 1799. He may also have preceeded both Steiner and Miquel in publishing the construction that leads to the Miquel Point. According to the Math History site at St Andrews, "Wallace also invented the pantograph, an instrument for duplicating a geometric shape at a reduced or enlarged scale."