If we begin with any triangle ABC and a point on the plane P, we can create the Pedal triangle for ABC and P. The vertices of the pedal triangle R, S, and T are the intersections on the perpendicular lines from P to the lines of ABC.
You may click HERE for the GSP script to create the pedal triangle for any triangle ABC and point P. Now we can look at the pedal triangles when P is moved about in the plane. The below drawing is when we choose P as the incenter of our triangle ABC.
Now lets look at the pedal triangle when P is the orthocenter, and we can look at when the orthocenter is outside the original triangle ABC. Notice when the orthocenter is outside the original triangle, the points D and E are reversed.
Now lets look at the pedal triangle point P is the circumcenter or the center of the nine-point circle of the triangle ABC.
We notice that when the circumcenter is outside of the original triangle ABC, the pedal triangle remains inscribed inside the triangle ABC, but like the orthocenter, the pedal triangle created when P is the center of the nine-point circle moves outside triangle ABC when the nine-point circle center is outside triangle ABC.
Now let's see what happens when P is on one of the sides of triangle ABC.
What if P was on one of the vertices of triangle ABC?
Our pedal triangle collapses into a line. There are many other possibilities for P, where our triangle will collapse into a Simson line, and they are all on the circumcircle.
Now if we locate the midpoints of the sides of the Pedal Triangle, and construct a circle with center at the circumcenter of triangle ABC such that the radius is larger than the radius of the circumcircle. We can then trace the locus of the midpoints of the sides of the Pedal Triangle as the Pedal Point P is animated around the circle we have constructed.
We can see that the three paths are ellipses. Lets see what happens if we repeat this process, but use the circumcircle instead of a circle with a larger radius.
We can see that when P is on the circumcircle the triangle collapses into the Simson line. And when the circumcircle is used as a path for P, the three ellipses formed when tracing the midpoints of the triangle have their endpoints on the vertices and segments across from the vertices.
Now we can choose P on the circumcircle so that the Simson line is the segment AC and then construct the orthocenter of triangle ABC.
We notice that the segment from P to the orthocenter intersects AC and the Simson line at their midpoint. This is also true if the Simson line is AB or BC.
If we create two pedal points on the circumcircle P1 and P2, we can connect the two Simson lines and examine the angle they form at intersection.
We can see that angle ABC is half of the angle formed by P1, P2, and the circumcenter.
Now lets put our P on the incircle, and animate P around the incircle and trace the midpoints of our legs of the pedal triangle.
We see that the locus of the midpoints are ellipses around the incenter. We also see that if ABC is a right triangle one of our ellipse becomes a circle.
We can also create an excircle to our triangle ABC, and see what the locus of the midpoints of the sides of the pedal triangle are when P moves along the excircle.
We can see that the locus of the midpoints of two of the side of the pedal triangle are ellipses, and the third is a circle.
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