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
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.