Assignment 9: Pedal Triangles
Given a triangle and an arbitrary point P, a pedal
triangle can be constructed.
The construction of the pedal triangle RST begins with constructing
perpendiculars to the sides of ABC from point P. The points there the
perpendicular lines intersect the sides of triangle ABC are the points
R,S, and T. Where R, S and T are the vertices of the pedal triangle.
Click on the image above to manipulate point P and the size/shape of
Now you ask what is the significance of the pedal
triangle with different positions of P with respect to the triangle
ABC. Let's look at the case when point P coincides with the orthocenter
of triangle ABC.
The figure above (Figure 2) shows
triangle ABC with its orthic triangle within (blue). Click on the image
to move point P so it coincides with point H (orthocenter of triangle
ABC). The pedal triangle will coincide with the orthic triangle when
the pedal point P coincides with point H. This is true even if triangle
ABC is obtuse.
Let's look at the case when point P coincides with the
of triangle ABC.
In the figure above (Figure 3) the point C is the circumcenter of
triangle ABC. Click on the image to move pedal point P over point C.
What does the pedal triangle seem to represent? If you said the medial
triangle you are correct. The figure below is an image of triangle ABC
with its medial triangle. Click on the image to confirm whether the
pedal triangle forms the medial triangle when the pedal point P
coincides with the circumcenter (C).
In fact, the pedal triangle does indeed
form the medial when the pedal point P coincides with the circumcenter
of triangle ABC. Even if triangle ABC is obtuse.