Simson Line
Begin with a triangle, ABC. Draw its circumcircle. Pick any point on the circle, D. Now, find the points on the lines AB, AC, and BC nearest to D. (That is, drop a perpendicular from D to these three lines—these lines are dotted in the figure below.) Those three points are colinear, and the line is known as the Simson line. In the applet below, you can move A, B, C, and D.