When the Pedal Point is the Orthocenter

by

Jane Yun

To construct a pedal triangle, let P be any point in the plane. Draw the perpendicular lines from the point P to the sides of ÆABC. If the point P is the outside the triangle, extend the sides of ÆABC. Then, let R, S, & T be the points of those intersection.

The triangle ÆRST is the pedal triangle associated with ÆABC and the point P.

Click here for the GSP script tool for the general construction of pedal triangle to ÆABC, where P can be dragged any point in the plane.

Let investigate when the pedal point P is the orthocenter, where it lies inside the triangle; that is, the triangle is not an obtuse. Recall that the orthocenter of a triangle is a single point where three altitudes meet.

We see that when the pedal point P is located at the
orthocenter, the pedal triangle, ÆRST, is called the orthic triangle or
altitude triangle - the triangle joining the feet of the altitudes of a
triangle. Also, the incenter of
the orthic triangle is the orthocenter of ÆABC.

Here
is proof that the orthocenter, H, of ÆABC is the incenter of the orthic
triangle, ÆRST.

The
partial part of pedal triangle lies outside the triangle. This is acceptable result since
the orthocenter lies outside the DABC.