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Summer Tuggle

Assignment 9

Explore the pedal triangle that is formed when P is the orthocenter.

A pedal triangle is formed when you project a point onto the sides of a triangle. You will drop a perpendicular from point P to each side of the triangle ABC in order to create points R, S, and T.  Connect the new points and triangle RST becomes the pedal triangle.  In this instance point P is the orthocenter of triangle ABC.

This special case when point P is the orthocenter creates triangle RST which is the orthic triangle of triangle ABC.  A property of orthic triangles is that the orthocenter of ABC is the incenter of RST.  More specifically, the altitudes of ABC are the angle bisectors of RST.

Since R, S, and T are created from the altitudes of the original triangle, the pedal triangle collapses when triangle ABC is a right or obtuse triangle.  In the case of right triangle ABC, the altitudes intersect at a vertex of the original triangle to form the right triangle.

This means that P, B, S, and R are all concurrent.  The line connecting the third point of the pedal triangle (T) to S and R is the angle bisector of the right triangle since we know that point P (which is also S and R in this case) is the incenter of triangle RST.

When triangle RST is obtuse two of the vertices of triangle RST appear on the extension of the respective sides of triangle ABC.  Therefore triangle RST is not contained completely inside of triangle ABC.