Assignment 9

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.