Pedal Triangles

LetŐs
begin by constructing triangle ABC and its pedal triangle for an arbitrary
point P, in the plane. The pedal triangle is formed by connecting the
intersections of the perpendicular lines from P to each of the sides of the
triangle.

Notice
that P is inside the triangle, letŐs see what happens if P is outside the
triangle.

It looks
as if when P is inside the triangle, then the pedal triangle is inside also.
Similarly, if P is outside the triangle then the pedal triangle will also be
outside. LetŐs continue to see if my conjecture is correct.

Now,
letŐs see what happens if P is a vertex of the triangle.

LetŐs try
the other vertices before I make a conjecture.

When P is
a vertex, the pedal triangle collapse. I wonder why?

What
about when P is on one of the sides of the triangle.

P then
becomes one of the vertices of the pedal triangle. This is true for all cases.

Now,
letŐs try some other locates for P.

What
about when P is the centroid?

The pedal
triangle appears to be equilateral. I checked to see if the vertices of the
pedal triangle intersect the midpoints of the triangle, it does not. But it was
pretty close.

What
about when P is the incenter?

I donŐt
see any special relationships with this. The pedal triangle is no longer
equaliteral and its vertices are not in any significant location. LetŐs try the
orthocenter.

The pedal
triangle becomes the orthic triangle. This is true because the orthocenter is
the intersection of the altitudes of triangle ABC. The three points used to
form the orthic triangle are feet of the altitudes. So the vertices of the
orthic triangle are formed from the perpendicular from the orthocenter to each
side. Since P is located at the orthocenter, the vertices of the pedal triangles
are the same as the vertices of the orthic triangle.

Lastly,
letŐs see what will happen if P lies on the circumcenter of the triangle.

The pedal
triangle becomes the medial triangle in this case. This is true because the
vertices of the medial triangle are the midpoints of the sides of the triangle.
The circumcenter is perpendicular to each side through its midpoint. Therefore,
when P is the circumcenter, the vertices of the pedal triangle would be the
midpoints of the sides of the triangle.