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Pedal Triangles

by

Ana Kuzle

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 Pedal Point P.

1. I used GSP to create a script tool for the general construction of a pedal triangle to triangle ABC where P is any point in the plane of ABC whether point P is outside the triangle

or inside the triangle.

2. What if pedal point P is the centroid of triangle ABC?

Well, nothing special. Pedal triangle is inside the triangle and P is the centroid.

3. What if pedal point P is the incenter of triangle ABC?

The incenter of a triangle is the point on the interior of the triangle that is equidistant from the three sides. It is the center of an inscribed circle. Thus, in this case the inscribed circle of the triangle ABC circumscribes its pedal triangle RST. P is incenter for triangle ABC and circumcenter for triangle RST.

4. What if pedal point P is the orthocenter of triangle ABC?

a) If point P is the orthocenter of an acute triangle ABC, then ABC's pedal triangle RST coincides with its orthic triangle.

b) If P is the orthocenter of a right triangle ABC, then points A, H, P, S and T coincide. In addition, ABC's pedal triangle RST becomes degenerated.

c) If P is the orthocenter of an obtuse triangle ABC, then its pedal triangle is party outside of the triangle. Likewise, pedal point P is outside the triangle since triangle ABC is obtuse.

5. What if pedal point P is the circumcenter of triangle ABC?

a) If the triangle ABC is acute triangle, we know that its circumcenter is inside the triangle. Therefore, its pedal triangle RST stays inside as well.

In addition, all the ''little'' triangles are congruent since TS, SR and RT are mid segments of BC, AB and CA, respectively and therefore parallel and equal to half of corresponding lengths of sides. Thus, triangle RST is similar to triangle ABC.

b) If triangle ABC is right triangle, then its circumcenter is the midpoint of hypotenuse. Thus, pedal point P is the midpoint of the hypotenuse and the pedal triangle RST is inside the triangle ABC.

c) If the triangle ABC is an obtuse triangle, then the circumcenter is outside the triangle. Thus, pedal point P is outside the triangle but the pedal triangle RST is inside the triangle ABC.

6. If point P is the center of a nine point circle of an acute triangle, then the pedal triangle RST is inside the triangle ABC as well as the pedal point P.

If point P is the center of a nine point circle, then both the pedal point P and the pedal triangle RST stay inside the triangle

or the pedal point P and part of pedal triangle RST are outside the triangle ABC.

7. a) If the point P is on any side of an acute triangle, then the pedal triangle RST stays inside the triangle ABC and one of the vertices of the pedal triangle RST coincides with the pedal point P.

b1) If the point P is on the leg of a right triangle, then the pedal triangle RST is inside the triangle. One of the vertex of the pedal triangle coincides with the vertex at right angle. Point P coincides with the vertex of a pedal triangle on that side.

b2) If the point P is on the hypotenuse of a right triangle, then the pedal triangle RST is inside the triangle ABC and is a right triangle. P coincides with the vertex of the triangle RST that is on the hypotenuse of triangle ABC. Triangles PRT and TBR are congruent since they share a common side TR, a right angle and PT=RB.

3. If the point P is on the side of an obtuse triangle, then the pedal triangle RST does not stay inside the triangle ABC. Pedal point P coincides with one of the vertex of the pedal triangle that is on the same side of the triangle ABC.

8. If pedal point P becomes one of the vertices, regardless of the triangle ABC being acute, right or obtuse, the pedal triangle RST degenerates. Thus, triangle RST becomes a line segment. Click here to see GSP file.

9. The three vertices of the Pedal triangle can be collinear. This line segment is called the Simpson Line. From my exploration (triangle RST was degenerated), the Simpson Line appears when the pedal point P is:

a) the orthocenter of a right triangle

b) one of the vertex of the triangle

c) point on the circumcircle of ant triangle (click here to see GSP file)

10. By tracing the locus of the midpoints of the sides of the Pedal Triangle as the Pedal Point P is animated around the circle such that the radius is larger than the radius of the circumcircle of the triangle ABC we get three ellipses. Click here to see the GSP file.

11. a) By tracing the locus of the midpoints of the sides of the Pedal Triangle as the Pedal Point P is animated around the circumcircle of the triangle ABC we get three ellipses. Click here to see the GSP file.

As mentioned in 9 c), pedal triangle RST is degenerated into Simpson line. Also, each of the ellipses coincides with the circumcircle. Each one of this points is one of the vertex of an ellipse.

See the previous one and notice the difference!

b) By tracing the Simpson line as the Pedal point is moved along different paths, we get the envelope of the Simpson line. That is, we get a hyperbolic triangle.

Click here to see GSP file.

12. By tracing the locus of the midpoints of the sides of the Pedal Triangle as the Pedal Point P is animated around the circle such that the radius is smaller than the radius of the circumcircle of the triangle ABC we get three ellipses. From the picture bellow we can see that each ellipse intersects the circle in exactly two points. Click here to see the GSP file.

13. Again, if we animate the Pedal point P about the incircle of ABC and trace the loci of the midpoints of the sides we get three ellipses.

a) If the triangle is acute triangle, the we get this.

Thus, all of the ellipses are inside the triangle. Each ellipse intersect the other two.

b) If the triangle is obtuse triangle, we get this.

Thus, one of the ellipses is partly outside the triangle.

c) If the triangle is a right triangle, we get this.

Thus, one of the ellipses is a special ellipse, circle!

d) If a triangle is equilateral triangle, we get this.

Thus, all of the ellipses are inside the triangle. They are congruent and the angle between them is equal to 120 degrees.

e) If the triangle is an isosceles triangle, we get this.

Thus, the major axis of the blue ellipse is the line of symmetry. That is, the green and the purple ellipses are congruent.

Click here to see GSP file for some of the cases mentioned above.