Sinje J. Butler

Pedal Triangles

The following is triangle ABC.  Point P is the pedal point (which could be any point in the plane).  From point P, lines have been constructed that are perpendicular to each side of the triangle.  These intersections are labeled R, S and T.  RST  is the pedal triangle.

Please click here for a GSP Demonstration that shows the pedal triangle for P at different places throughout the plane.

The following will explore some correlations that can be made between the pedal triangle and others triangles dependent upon the location of P.

If pedal point P is the orthocenter of triangle ABC the pedal triangle is the orthic triangle.  This is true for the case when the orthocenter is inside triangle ABC and this is the case when the orthocenter is outside the triangle.  Please click here for a GSP Demonstration.

This occurs because the orthocenter of ABC lies on each of the three altitudes of this triangle.  When P is the orthocenter it also lies on each of the three atlitudes.  Therefore, P is perpindicular to each side at the vertices of the orthic triangle and thus the orthic triangle is the pedal triangle.

If the pedal point P becomes the circumcenter,  the pedal triangle is the medial triangle.  This occurs because when P is placed on the circumcenter it lies on the perpendicular bisectors of the sides of triangle ABC.  So R, S and T are perpendicular to the original triangle and they lie on the three midpoints of the three sides, thus when these segments are connected they are the medial triangle.  Please click here for a GSP Demonstration of P as the circumcenter. Notice, that if the circumcenter is outside of the triangle ABC, when P lies on top of the circumcenter, it is the medial triangle.

Other Observations

Please click here for GSP Demonstration to move P throughout the plane.(PedCircmcir1)  Notice that as P is moved around it appears there are positions of P for which the vertices of the pedal triangle are collinear.  Where does this occur and why?

The next GSP Demonstration  appears to show that if the circumcirle of triangle ABC is constructed and point P is merged onto the circumcircle, the vertices of the pedal triangle are collinear for any point on the circumcircle.(PedCircmcir2)