Exploring
Pedal triangles

By
Princess Browne

To
construct a pedal triangle, we will construct any triangle ABC and a point P in
the plane of that triangle. We will construct line segments through P and
perpendicular to each sides of the triangle. The three points X,Y, and Z formed
by the intersection of the perpendicular lines give us the pedal triangle.

To construct another pedal
triangle similar to the original triangle, we will use the pedal triangle
within the pedal triangle. For example, we will used the pedal triangle XYZ to
find the pedal triangle of RST, and then use pedal triangle of RST to find the
pedal triangle of AÕBÕCÕ.

We
will use (GSP) GeometerÕs Sketchpad to show that triangle ABC and pedal
triangle of AÕBÕC are similar by calculating the ratio of the sides of both
triangle.

AB =
10.57cm

AC =
11.45cm

BC =
12.86cm

AÕBÕ
= 1.22cm

AÕCÕ
= 1.32cm

CÕBÕ
= 1.49cm

AB/AÕBÕ
= 8.66

BC/BÕCÕ
=8.66

AC/AÕCÕ
= 8.66

As we
can see that the corresponding sides have the same ratio therefore the original
triangle is similar to the pedal triangle AÕBÕC.

**What
happens if the pedal point is outside the original triangle?**

For
the obtuse triangle below, we can see that when the pedal triangle in inside
triangle ABC, the pedal point is outside of triangle ABC. When P moves towards any
of the vertexs on triangle ABC, the pedal triangle is degenerate.

Click
here to explore with the pedal triangle