Pedal Triangles

By Courtney Cody

In this assignment, we will construct a pedal triangle where the pedal point P is any point in the plane.  Then we will explore the case when P is the pedal point P is the centroid, circumcenter, incenter, orthocenter, and center of the nine-point circle of a given triangle.

Suppose we have an arbitrary triangle ABC and let P be any point on the plane.  By constructing the perpendicular lines from each side of triangle ABC to point P and calling the intersections of these lines points R, S, and T, we have constructed a pedal triangle.  Namely, the triangle formed by points R, S, and T is the pedal triangle and P is the pedal point.  In this case, we have P lying outside of triangle ABC.

Click on the diagram above to move vertices of triangle ABC and the pedal point P and observe how the pedal triangle RST changes as these points are moved.

Now we will look at the cases when the pedal point P is the centroid of triangle ABC.  By definition, the centroid is the common intersection of the three medians.  By constructing the centroid of triangle ABC and letting it be the pedal point P, we can construct the pedal triangle about P.

If we study the diagram above, we see that when the pedal point P is the centroid of triangle ABC, the pedal triangle RST is always inside triangle ABC since the centroid is always inside triangle ABC.  Click on the diagram above to move vertices of triangle ABC and the pedal point P and observe how the pedal triangle RST changes as these points are moved when P is the centroid.

Now we will look at the cases when the pedal point P is the circumcenter of triangle ABC.  By definition, the circumcenter is the intersection of a triangleÕs three perpendicular bisectors. By constructing the circumcenter of triangle ABC and letting it be the pedal point P, we can construct the pedal triangle about P.

If we study the diagram above, we see that when the pedal point P is the circumcenter of triangle ABC, and that the pedal triangle RST is actually the medial triangle since itÕs vertices are the midpoints of the segments that make up triangle ABC.  Additionally, since P is the center of the circle that circumscribes triangle ABC then P is equidistant from all three vertices.  This implies that the distance from P to each of the vertices A, B, and C is the radius of the circles that circumscribes triangle ABC.  Click on the diagram above to move vertices of triangle ABC and the pedal point P and observe how the pedal triangle RST changes as these points are moved when P is the circumcenter.

Now we will look at the cases when the pedal point P is the incenter of triangle ABC.  By definition, the incenter is the common intersection of the three angle bisectors.  By constructing the incenter of triangle ABC and letting it be the pedal point P, we can construct the pedal triangle about P.

If we study the diagram above, we see that when the pedal point P is the incenter of triangle ABC, pedal triangle RST will also always be inside triangle ABC since the pedal point P is the center of the circle that is inscribed in triangle ABC.  Hence, P is always inside triangle ABC.  Additionally, since P is the center of the circle that is inscribed by triangle ABC then P is equidistant from all three vertices of the pedal triangle RST. This implies that the distance from P to each of the vertices R, S, and T is the radius of the circles inscribed by triangle ABC.  Click on the diagram above to move vertices of triangle ABC and the pedal point P and observe how the pedal triangle RST changes as these points are moved when P is the incenter.

Now we will look at the cases when the pedal point P is the orthocenter of triangle ABC.  In particular, weÕll start by looking at an acute triangle ABC.  By definition, the orthocenter is the intersection of the three altitudes of a triangle.  By constructing the orthocenter of acute triangle ABC and letting it be the pedal point P, we can construct the pedal triangle about P as pictured below:

When the pedal point P is the orthocenter of triangle ABC, pedal triangle RST is also the orthic triangle, which is defined as the triangle joining the feet of the altitudes of a triangle.  Since altitudes always meet the sides of the larger triangle (ABC in this case), then this is how we found the pedal triangle since it the vertices of RST are where the lines perpendicular to the sides of ABC pass through P.  Additionally, the pedal triangle is located entirely within triangle ABC.  Click on the diagram above to move vertices of triangle ABC and the pedal point P and observe how the pedal triangle RST changes when P is the orthocenter of an acute triangle.

What would happen if triangle ABC were obtuse?

If triangle ABC were obtuse, then we must extend the sides of the triangle in order to determine where the altitudes intersect the sides of the triangle.  Once we have constructed our pedal triangle RST, we notice that it is located partially inside triangle ABC and partially outside.  Also, our pedal point P is located outside of triangle ABC.  Hence, the pedal point is called the exterior orthocenter.  Click on the diagram above to move vertices of triangle ABC and the pedal point P and observe how the pedal triangle RST changes when P is the exterior orthocenter.

What about if triangle ABC were a right triangle?

If triangle ABC were a right triangle, then we notice that points A, P, R, and S all converge.  This is because when we construct the altitudes of each vertex in order to find the orthocenter, two of the altitudes lie directly on the two sides adjacent to the right angle.  The third altitude actually bisects triangle ABC.  Hence, the orthocenter of a right triangle is located at the vertex of triangle ABC.  Specifically, P is the vertex of the right angle.  Now, if we construct the pedal triangle, since points P, R, and S lie on the same point, then the only segment drawn for the pedal triangle is segment PT, which is the line that bisects triangle ABC.  Thus, the pedal triangle is a line on an altitude of triangle ABC.  Click on the diagram above to move vertices of triangle ABC and the pedal point P and observe how the pedal triangle RST changes when P is the orthocenter of right triangle ABC.