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 locate three points R, S, and T that are the intersections. Triangle RST is the PEDAL TRIANGLE for PEDAL POINT P.
Now we will have a script of that here.
Now we are going to examine what happens if our pedal point is in particular spots.
If the pedal point is the centroid of the triange we get:
If the pedal point is the incenter we get:
If we make P the orthocenter we get:
Even if the orthocenter is outside-(NOTE: the pedal triangle is the orthic triangle when it is inside)
When P is the circumcenter of the triangle:
Even when the circumcenter is outside the triangle as so:
When P is the center of the nine point circle:
When P is on a side of the triangle:
When P is one of the vertices of the triangle:
There are some cases in which three vertices of the Pedal triangle are colinear. This segment is then called the Simson Line.
Locating the midpoints of the sides of the Pedal Triangle and constructing a circle with center at the circumcenter of triangle ABC such that the radius is larger than the radius of the circumcirle and tracing the midpoints of the sides of the Pedal triangle as Point P is animated around the circle gives you three paths. Click here
Repeating this with the path being on the circumcirle also produces three paths. Click here
When the lines of the Pedal Triangle are traced, rather than the midponts we can then experiment with different paths and those figures produced. In particular we want to examine the circumcircle. Click here.
Now we are going to do the same process only now our outside circle is less than our circumcircle and look how it has changed, click here.
Other examinations can be made...what can you come up with?