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Assignment 9

Pedal Triangles

by Emily Bradley

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Prove the pedal triangle of the pedal triangle of the pedal triangle of a point is similar to the original triangle. That is, show that the pedal triangle A'B'C' of pedal triangle RST of the pedal triangle XYZ of pedal point P is similar to triangle ABC.

 



First we establish the property shown in the above diagram.

P is the circumcenter of triangle ABC and triangle XYZ by definition of pedal triangles. A circle can be drawn where A, Z, P,and Y lie on the circumference. This can be done on all vertices of the original triangle. Notice that angle ZAP and angle ZAP subtend the same arc of the circle that their vertices lie on.

Therefore

This holds true for all vertices and each pedal triangle of a triangle.

 

 



To prove that the third pedal triangle is similar to a given triangle show that corresponding angles are congruent.

We want to begin by showing that

Breaking down angle BAC as the sum of two angles:



 

 



Each of these angles has the congruence property that we established first.



 

And the other angle