Mathematics Education
Last modified on June 5, 2009.
1a. 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.
Example, when Pedal Point is outside the triangle
Example when Pedal Point is inside the triangle
Click HERE to download a GSP file.
Drag the Pedal Point P to various locations.
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.
2. What if pedal point P is the centroid of
triangle ABC?
3. What if . . . P is the incenter . . . ?
4. What if . . . P is the Orthocenter . . . ? Even if outside
ABC?
5. What if . . . P is the Circumcenter . . . ?
Even if outside ABC?
6. What if . . . P is the Center of the nine point
circle for triangle ABC?
7. What if P is on a side of the triangle?
8. What if P is one of the vertices of triangle
ABC?
9. Find all conditions in which the three vertices
of the Pedal triangle are colinear (that is, it is a degenerate
triangle). This line is called the Simson Line.
10. 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?
11. Repeat where the path is the circumcircle.
11a. Construct lines (not segments) on the sides
of the Pedal triangle. Trace the lines as the Pedal point is moved
along different paths.
11b. 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.
12. Repeat where the path is
a circle with center at the circumcenter but radius less
than the radius of the circumcircle.
13. Is there a point on the circumcircle for P
that has side AC as its Simson line? AB? BC?
14. 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.
15. 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.
16. 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.
17. 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?
18. 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 specials cases for right triangles, isosceles
triangles, or equilateral triangles?
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