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Rayen Antillanca. Assignment 9


Let ABC be any triangle, and let P be any point on the plane. From P construct perpendiculars lines to the sides of triangle ABC. Let R, S, and T be the intersection points between the sides of the triangle ABC and the perpendicular lines from P, as shown in figure below. Triangle RST is the Pedal Triangle for Pedal Point P.

The triangle RST is the pedal triangle. Point P is outside of triangle ABC

Pout.gif

 

The triangle RST is the pedal triangle. Point P is inside of the triangle ABC

Pin.gif

 

If you want to draw a pedal triangle, you can use this GSP tool.

 

 

Prove the pedal triangle of a pedal triangle of a 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 the triangle ABC.

pedal.gif

 

 

Step 1

ped1.gif

Let C be the circle with AP as a diameter. Since both angle PXA and angle PZA are right

angles and both subtend AP, points X and Z lie on circle C. Since both angle PAZ and angle PXZ are on C and subtend the segment PZ, angle PAB = PXZ.

 

 

Step 2

Now I am going to prove that the pedal AíBíCí is similar to ABC triangle.

ped2.gif

Let = angle PAB. So by step 1 I have

Also letting

So

 

The procedure is the same for the other angles. So, for criteria AAA triangle AíBíCí is similar to triangle ABC

 

 

 

Note: All figures on this web page were made with The Geometerís Sketchpad

 

 

 

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