This activity investigates the
properties of pedal triangles.
Using GSP, create any triangle ABC and any point P in the plane.
Construct perpendicular lines to the sides of ABC through point
P. Label the three intersections formed by these perpendiculars
and label them D, E, F. Connect D, E, F using line segments to
form a new triangle. Triangle DEF is the pedal triangle for pedal
point P.
We will now examine several
special cases of pedal triangles.
Remember that the incenter of a
triangle in the intersection of the three angle bisectors of the
triangle. Let's examine what happens when we make the
incenter of the triangle pedal point P.
Does
this particular triangle have any special properties? Actually
pedal point P is both in incenter of the original triangle as well as
the circumcenter of the pedal triangle.
Notice
how the incenter and pedal point (P) is also the intersection of the
angle bisectors of pedal triangle DEF, also known as the
circumcenter.
This property holds for all triangles.
Let's look at this property more in
depth.
Consider the incircle of triangle ABC. To find the radius
of this circle, drop a perpendicular from the incenter to a side of the
circle. The intersection will also be one of the vertices of the
pedal triangle.
Remember that the circumcenter is the intersection of the perpendicular
bisectors of the angles of a triangle. Let's observe what happens
if pedal point P is the circumcenter of triangle ABC.
The pedal triangle formed is the medial
triangle of triangle ABC.
Click here for an interactive javasketchpad
exploration. Move points A, B, and C. What happens to
the pedal triangle? What happens to pedal point P? Notice
how this property still holds even when point P is outside triangle
ABC.
