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Mathematics Education

EMAT 6680, Professor Wilson

Exploration Number 9, Pedal Triangles by Ursula Kirk

 

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 ABC of pedal triangle RST of the pedal triangle XYZ of pedal point P is similar to ABC.

STEP 1

By definition of pedal triangle and as shown in the diagram, the pedal point is the circumcenter of triangle XYZ, and it is also the circumcenter of triangle RST. The circumcenter of a triangle is the point where the perpendicular bisectors of the sides intersect.

 

pcircumcenter.JPG

 

pcircumcenter2.JPG

 

We can draw a circle O which passes trough points YRT and the pedal point. This can also be done on vertex Z and on vertex X. Then the angle formed by points RY and the pedal triangle is the same that the angle formed by points R, the pedal point and T.

pedaltriangleSTEP1.JPG

Now, we must show that the third pedal triangle is similar to a given triangle by showing that the corresponding angles are congruent. That is, we have to show that angle ABC is congruent to angle ABC. We can rewrite angle ABC as BAPedal+PedalAC which is congruent to ZAPedal+PedalAY.

 

pedaltriangleSTEP2.JPG

Then

Also

And

Therefore

Similarly

 

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