EMAT 6680 - ASSIGNMENT 9 - PEDAL TRIANGLES

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.

Q1. 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.

Q2. What if pedal point P is the centroid of triangle ABC?

Q3. What if pedal point P is the incenter of triangle ABC?

Q4. What if pedal point P is the orthocenter of triangle ABC? There are three cases:

Click to download the animation file that comprises all three cases we studied above:

What happens if we use the same animation file with traces enabled for pedal triangle RST? We get a UFO with two arms extending on the right and on the left to infinitiy.

Q5. What if pedal point P is the circumcenter of triangle ABC? Answer: Pedal and medial triangles coincide: They are congruent. There are three cases:

Q6. What if pedal point P is the center of the nine point circle for triangle ABC? There are two cases:

Q7. What if P is on a side of the triangle? There are three cases:

Q8. What if P is one of the vertices of triangle ABC? Answer: We have only one case irrespective of whether ABC is acute, right, or obtuse: The pedal triangle RST is degenerate: IT'S NOT A TRIANGLE! IT'S ALWAYS A LINE SEGMENT!

Q9. Find all conditions in which the three vertices of the Pedal triangle are colinear (that is, it is a degenerate triangle). This line segment is called the Simson Line.

From what I did above, it looks like Simson Line appears when the pedal point is: (1) on one of the vertices of any triangle, (2) the orthocenter of a right triangle, in particular, (3) on the circumcircle of any triangle.

Q10. 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?

Q11. Repeat where the path is the circumcircle? Click below to download this GSP file. I liked it very much.

Q12. Construct lines (not segments) on the sides of the Pedal triangle. Trace the lines as the Pedal point is moved along different paths. 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.

Q13. Repeat where the path is a circle with center at the circumcenter but radius less than the radius of the circumcircle.

Q16. 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.