Given: P lies on side AC and triangle ABC is a right triangle.
Show: Triangle RTS, the pedal triangle is a right triangle.
Note: The reasons follow the statements in parantheses and are purple.
If triangle ABC is a right triangle, then angle B is 90 degrees (by def of right triangle), and side BC is perpendicular to side AB (by def of perpendicular). We know that line PR is perpendicular to side AB (by def of pedal triangle: Recall that R is the intersection of side AB and the line perpendicular to AB from point P). Therefore, we can say that PR and BC are parallel (Thm. If two lines are perpendicular to the same line, then the two lines must be parallel). We also know that line PS is perpendicular to side BC (again by the def of pedal triangle). So angle PSC is 90 degrees. We then know that angle SPR is 90 degrees (Line PS is a transversal through parallel lines RP and BC, which form alternate interior angles SPR and PSC, and alternate interior angles are congruent.). So triangle RPS is a right triangle. This implies that triangle RTS is a right triangle, since P and T are concurrent (follows from proof of observation #1).