Given: Point P lies on a side (extended if necessary) of triangle ABC.

Show: Point P is concurrent with vertex of pedal triangle.

Recall: The Pedal triangle is formed by the intersection of the lines perpendicular to the sides of triangle ABC and the sides (extended if necessary) of triangle ABC.

i.e. line PS is perpendicular to line BC, and S is the point of intersection of PS and BC.

From the given, P lies on line BC, but P also lies on the line perpendicular to line BC from point P (by def of pedal triangle). Thus, P is the intersection of line BC and the line perpendicular to BC from point P (two lines intersect at a point). Therefore, since there is one unique perpendicular line from point P to side BC, then P must be concurrent with S.