The pedal triangle, RST, is the triangle formed by choosing any arbitrary point, P (the pedal point) and any arbitrary triangle ABC. Construct the lines perpindicular to each side of triangle ABC and through P. The points of intersection (R, S, and T) of these perpindicular lines and the sides of triangle ABC form the pedal triangle, triangle RST. Click here for a GSP sketch pad in order to create your own pedal triangle.
In the following
page, we will look at various pedal triangles constructed by allowing
the pedal point to be specific points related to the triangle
ABC. We will also explore other characteristics of the pedal triangle
Now let's allow the Pedal point to be the INCENTER of the triangle ABC. The incenter of a triangle is the point inside of a triangle that is equidistant from its sides. The incircle, therefore, goes through the points of intersection of the lines through the incenter and perpindicular to each side of triangle ABC. These points of intersection are the vertices of the pedal triangle, R, S and T, by definition of the pedal triangle.
circumcenter is the intersection of the perpindicular bisectors
of triangle ABC, the midpoints of triangle ABC become the vertices
of the pedal triangle when we allow the circumcenter to be the
1. What type of objects do you think would be formed if we trace the midpoints of the legs of the pedal triangle as the pedal point, P, follows the path of the circumcircle of triangle ABC? Click here for a Geometer's Sketch Pad animation demonstrating this.
2. What if we traced the midpoints of the legs of the pedal triangle again, but this time P followed the path of a circle centered at the circumcenter but with a radius larger than that of the circumcircle? Can you predict the types of figures traced out? Click here to see the results.
3. If we let
the pedal point, P, follow the path of the circumcircle but as
it did so, we traced the lines whose intersections make the pedal
triangle, we obtain a single, special figure. Can you guess what
this figure will be? Click here to see.
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