# EMAT 4680/6680 Explorations 09 Pedal Triangles

1a. Let triangle ABC be any triangle. Then if P is any point in the plane, construct 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.

Example, when Pedal Point is outside the triangle

Example when Pedal Point is inside the triangle

1b. Use 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.

1c. Prove the pedal triangle of the pedal triangle of the pedal triangle of a point is similar to the original triangle. That is, show that the pedal triangle A'B'C' of pedal triangle RST of the pedal triangle XYZ of pedal point P is similar to triangle ABC.

This is one of the fundamental theorems about Pedal Triangles.

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

2b.  What if . . . P is the incenter . . . ?  Then the pedal triangle has its vertices at the tangent points of the incircle.   Prove that the three segments from the vertices of triangle ABC to the tangent points of the incircle on the opposite side are concurrent.    This point of concurrency is the Gergonne Point of triangle ABC.  The triangle formed by the points of tangency of the incircle is called the Intouch Triangle or the Contact Triangle.

2c.  What if . . . P is the Orthocenter . . . ?   Prove that the Pedal triangle is the Orthic triangle of the original triangle.    Even if the Orthocenter outside ABC?

2d.  What if . . . P is the Circumcenter . . . ? Prove that the Pedal Triangle is the Medial Triangle of the original triangle.        Even if the circumcenter is outside ABC?

2e.  What if . . . P is the Center of the nine point circle for triangle ABC?

3a.  What if P is on a side of the triangle?

3b.  What if P is one of the vertices of triangle ABC?

4.  Find all conditions in which the three vertices of the Pedal Triangle are colinear (that is, the Pedal Triangle is a degenerate triangle). This line is called the Simson Line.

Prove:     If  P is any point on the circumcircle of triangle ABC, the the feet of the perpendiculars from P to the sides of the triangle (possible extended) are colinear.

5.  Locate the midpoints of the sides of the Pedal Triangle. Construct a circle with center at the circumcenter of triangle ABC such that the radius is larger than the radius of the circumcircle. Trace the locus of the midpoints of the sides of the Pedal Triangle as the Pedal Point P is animated around the circle you have constructed. What are the three paths?

6. Repeat #5 where the path is the circumcircle.

7a. Construct lines (not segments) on the sides of the Pedal triangle. Trace the lines as the Pedal point is moved along different paths.

7b. In particular, find the envelope of the Simson line as the Pedal point is moved along the circumcircle. Note, you will need to trace the image of the line, not the segment.

The resulting outline is the Deltoid Curve.

7c.   Repeat where the path is a circle with center at the circumcenter but radius less than the radius of the circumcircle.

8. Is there a point on the circumcircle for P that has side AC as its Simson line? AB? BC?

9. Construct the Simson line of a point P (i.e. put P on the circumcircle) and construct the segment connecting P to the Orthocenter. Trace the locus of the intersection of the Simson Line and the segment connecting the Orthocenter to the Pedal Point as the Pedal point is moved around the circumcircle. Prove that the locus is the nine-point-circle of ABC.

10. Select two pedal points on the circumcircle and construct their Simson lines. Compare the angle of intersection of the two Simson lines with the angular measure of the arc between the two pedal points.

11. Animate the Pedal point P about the incircle of ABC. Trace the loci of the midpoints of the sides. What curves result? Repeat if ABC is a right triangle.

12. Construct an excircle of triangle ABC. Animate the Pedal point P about the excircle and trace the loci of the midpoints of the sides of the Pedal triangle. What curves result? Look at the angle bisectors through the center of the excircle. How are the loci positioned with respect to the angle bisectors?

13. Other investigations. Have you found any observations about pedal points and pedal triangles other than the ones from the previous suggestions? If so, discuss them. If not, try something else. What about special cases for right triangles, isosceles triangles, or equilateral triangles?