Assignment 9
Page Bird
Pedal
Triangles
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.
Click here for a GSP script tool
for the construction of a pedal triangle.
When the
centroid or the incenter is the P point of the pedal triangle, the pedal
triangle always lies within the given triangle ABC. Here is one such example.
To see a GSP in which the centroid and the Pedal Point P are the same click here. To see a GSP file in which the incenter and the Pedal Point P are the same click here.
There are several cases when the orthocenter
is pedal point P.
Case 1: Acute triangle ABC
The pedal triangle RST
lies within ABC as seen below
Case 2: ABC is a right
triangle
When ABC is a
right triangle then H or P becomes a vertex, and triangle RST collapses to a
straight line as seen below.
Case three: Triangle ABC is an obtuse triangle
When ABC is an obtuse triangle, the orthocenter is located outside triangle
ABC and part of triangle RST is also outside of triangle ABC. See below:
Click here to use a GSP
file with the orthocenter constructed as the pedal point P. Just drag on A, B, or C to change the
triangle and location of pedal point P or H.
When pedal point P is the
circumcenter, the pedal triangle always lies inside triangle ABC, regardless of
where the pedal point P lies.
Click here to check
it out.
When P lies on a vertex, triangle RST collapses to
the altitude 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 segment is called the Simson
Line.
The
example above where P lies on the vertex, is one example of the Simson Line.
Anytime a vertex of pedal triangle RST becomes the same as a vertex of
triangle ABC then the pedal triangle degenerates.
The best
way to see this is by dragging point P until the vertices line up for the
triangles.
Click here to investigate on your own.
One of the most interesting parts
of the assignment came in the later explorations:
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?
As you can
see the path of the locus for the midpoints are ellipses. I had a problem constructing this
because I initially had triangle RST limited by triangle ABC, so I was only
getting a partial picture of the ellipses. Click here to see a GSP file of this.
11. Repeat where the path is
the circumcircle?
As you can see the paths are
still ellipses, just fatter than for a larger circle.
Click here to see a GSP file of this picture.
11a.
Construct lines (not segments) on the sides of the Pedal triangle. Trace the
lines as the Pedal point is moved along different paths.
Tracing the
Simson line as the pedal point was animated along the circumcircle was a neat
image:
The beauty
of GSP is that it makes visible geometric relationships that we would
otherwise, in most cases, be blind to.
Even the invisible is ordered, and that is why this activity is so cool! I think any student would be interested
in this investigation.
My next
investigation would start with the question, so what are the foci of the
ellipses, which are the locus of the midpoints of the sides of triangle
RST?