A cool fact about pedal triangles and its proof

(I worked out the proof on Oct.25, 2009 and technologized it the second day; it would be my 7th finished write up.)

Chen Tian

Let triangle ABC be any triangle. Take P any point in the plane; then the triangle formed by constructing, through P, perpendiculars to the sides of ABC (extended if necessary) locate three points X, Y, and Z that are the intersections. Triangle XYZ is the Pedal Triangle for Pedal Point P.

Here is the pedal triangle script tool I created. You will see that the pedal point can be on, or inside, outside the triangle.

Interestingly, if we construct a pedal triangle inside a previous one continuously, we guess there must be some connection between some of them. Now let's prove the cool fact that if the third pedal triangle is similar to the original triangle (we only consider the case when the pedal point is inside the original triangle, since the outside case would be similar).

We really need a big figure to see it clearly.

Note when the pedal point is on the circum-circle of the original trianlge those pedal points will be colinear (that is, it is a degenerate triangle), and this line is called the Simson Line (this would be another good exploration and proof for students to put their hands on, but I won't do it here since it is getting to the end of the semester having crazy schedule). But the idea of the proof for Simson Line is similar to the above one, i.e., four points are on a circle if two pairs of segemnts connecting the four points form two right angles; and also the vertical angles are equal to each other.