Pedal Triangles

Write-up by

Blair T. Dietrich

EMAT 6680

Assignment #9

This investigation will focus on pedal triangles. Let triangle ABC be any triangle. Then if P is any point in the plane, then
the triangle formed by constructing perpendiculars to the sides of ABC
(extended if necessary) locate three points R, S, and T that are the
intersections. Triangle RST is the ** Pedal
Triangle** for

If P is the centroid of triangle ABC, the pedal triangle appears to be the medial triangle but, in fact, it is not.

Shown below are two graphs. The first one shows P coinciding with the centroid G. In this picture, it can be seen that the points R, S, and T are not the midpoints of segments AB, BC, and CA, respectively.

The second one shows the points R, S, and T coinciding with the midpoints of segments AB, BC, and CA, respectively, and that P is clearly not the same as G.

Notice that sometimes the Pedal Triangle is in the interior
of triangle ABC and sometimes it is not.
Look at **this animation** and try to decide how the
position of P affects the location of the Pedal Triangle. You can also stop the animation and drag the
point P yourself.

When the Pedal Point is moved along the circumcircle, the locus of the Simson Line forms a triangular-shaped region "under stress" called a deltoid.

Try this one yourself.
**Click here.**

What happens if the midpoints of the sides of the Pedal Triangle are traced? In the graph below, the pedal point P was allowed to circumnavigate the circumcircle. Notice that the midpoints trace out three ellipses.

Notice that when triangle ABC is a right triangle, one of the midpoints traces out the path of a circle rather than an ellipse.

To observe an animation of the traced midpoints of the sides
of the Pedal Triangle, **click here.**