Pedal Point Match
Christa Marie Nathe
The construction of a pedal triangle is created with respect to a pedal point. Take any triangle and label it ABC and pick any point on the plane that will be the pedal point of the triangle. This point, P, can either be inside or outside the triangle. Next drop perpendicular lines from P to the three sides of the triangle. It may be necessary to extend the lines of the triangle to the extent that they will intersect the orthogonal lines dropped from point P. The points of intersection of the perpendicular lines with the sides (or extended sides) of the given triangle will form the vertices of the pedal triangle.
Below you can observe some manipulations of triangle ABC, the pedal triangle with respect to the pedal point P.
In the following investigation we will observe what happens to the pedal triangle when the pedal point is located at the centroid, incenter, orthocenter and circumcenter of triangle ABC.
Recall that the centroid of a triangle is created by the intersection of the three medians, or the intersection of the segment from a vertex to the midpoint of the opposite side.
Now letŐs move the pedal point P into a position of one of the vertices of the triangle to observe what happens to the pedal triangle. As you can observe, the pedal triangle collapses when P is located at a vertex.
When the pedal point is matched to the centroid, the resulting pedal triangle is a medial triangle.
The creation of the incenter of a triangle is a point in the interior that is equidistant from the three sides. The incenter point lies on the angle bisector of each angle of the triangle.
When the pedal point P is matched with the position of the incenter the pedal triangle becomes the medial triangle.
Constructing the orthocenter of a triangle is as easy as finding the intersection of the three lines containing the orthogonal segment from the vertex to the line of the opposite side of the triangle.
When the pedal point is matched with the orthocenter, then the pedal triangle will have a perpendicular relationship to the sides of the ABC triangle.
When the pedal point is located anywhere on the circle, the pedal triangle collapses.
When the pedal point and circumcenter match, it yields a medial triangle.