Erik D. Jacobson

Pedal Triangle Similarity
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In this exploration, I prove that the pedal triangle of the pedal triangle of the pedal triangle of a point P is similar to the original triangle ABC. That is, I demonstrate that show that the pedal triangle TUV of pedal triangle QRS of the pedal triangle RST of pedal point P is similar to triangle ABC (figure 1).

*Figure 1: The pedal
triangle of a pedal triangle of a pedal triangle of point P and triangle ABC.*

The first step is to observe that the circumcircle of triangle PST passes through point A (a fact that is easily follows from the definition of pedal triangles). Next, observe that the circumcircle of triangle with vertices P, X, and the intersection of line TU and line PS (call this unlabeled point Y from now on) passes through point T, one vertex of the final pedal triangle TUV. What is important to realize is that the quadrilaterals ATPS and TXPY share angle SPT which is supplementary to angle BAC and YTX. Thus angle A in triangle ABC is congruent to angle T in triangle TUV (see Figure 2).

*Figure 2: Angle SPT is
supplementary to angle A in triangle ABC and angle T in triangle TUV.*

A similar argument can be made after noting that the circumcircle of triangle PRT passes through point C. Next, the circumcircle of PWX (where X and W are the intersections of the extended sides of triangle TUV) passes through point U (see Figure 3). Now angle RPT is supplementary to angle RCT and angle WUX, thus angle RPT is congruent to angle ACB and to angle TUV. The transitive property gives that angle ACB is congruent to angle TUV. We conclude that triangle ABC is similar to triangle TUV by the theorem of Angle-Angle Similarity.

*Figure 3: Angle RPT is
supplementary to angle RCT and angle WUX, thus angle RPT is congruent to angle
ACB and to angle TUV.*