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 A’B’C’ 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.
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.
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 A’B’C’. We can rewrite angle ABC as BAPedal+PedalAC which is
congruent to ZAPedal+PedalAY.
Then
Also
And
Therefore
Similarly
