Janet Kaplan

PEDAL TRIANGLES

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Let triangle ABC be any
triangle. Construct a point P anywhere in the plane, and construct
perpendicular lines from P to each side of ABC. Locate points R, S and T as the
intersections of the sides of ABC and the perpendicular lines. Extend the lines
of ABC in order to do so, if necessary.

Triangle RST is the **Pedal
Triangle** for **Pedal Point P**. Note that P can be anywhere in the
plane, either inside or outside of ABC.

CLICK HERE to see an initial Pedal Triangle construction which can be
manipulated.

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As you move Pedal Point P
all around, and adjust the size and shape of triangle ABC, you will notice that
the Pedal Triangle occassionally degenerates into a straight line segment. That
is, all three vertices of RST become collinear. When this occurs, it is known
as the **Simson Line. **What are the conditions under which this happens?

It looks as if Point P has
to be fairly close in towards the sides of ABC, and it definitely occurs when
Point P lands on any of its vertices. Any conjectures?

In fact, In 1797 William
Wallace proposed the following theorem. "The pedal triangle of a point P
with respect to triangle ABC degenerates into a straight line if and only if P
lies on the **circumcircle **of ABC." R, S and T are collinear only
when P is on the circumcircle, and P is on the circumcircle only when R, S and
T are collinear.

CLICK HERE for a GSP demonstration of this fact.

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We're going to look a
little further into the patterns that are formed by pedal triangles.

Locate the midpoints of the
sides of the Pedal triangle RST. Construct the **circumcircle** of triangle
ABC, that is, a circle with the center at the **circumcenter** of triangle
ABC and arc at the vertices. Now merge Point P onto this circumcircle. Trace
the paths of the three midpoints as you animate Point P around the
circumcircle. What are the paths?

CLICK HERE to see the animation.

The paths are ellipses in
which the major axis is along the same vector as the angle bisector of each
vertex of triangle ABC! And the length of each major axis is equal to each of
the altitudes of ABC. Interesting! This would be a point of further study for
me. I'd like to explore the mathematics behind this phenomenon.

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This next picture I call
"the hammock". It is, in fact, the envelope of the Simson Line as the
pedal point P is moved around the circumcircle. Construct lines over the line
segments forming the sides of triangle RST, and mark them for tracing. Then
animate point P again for its journey around the circumcircle. It makes a
beautiful picture……CLICK HERE to see the animation in progress.

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