Third Pedal Triangle
By Thuy Nguyen
Theorem: The third pedal triangle is similar to the original triangle.
Click here for the GSP script for a general construction of a pedal triangle ABC where P is any point in the plane ABC.
Click here for the GSP script to see the third pedal triangle.
We first note that P lies on the circumcircles of all the triangles AB1C1, A2B1C2, A2B2C1, and A3B2C3. Now join P to every vertex on all four triangles. Then we have
<C1AP = <C1B1P = <A2B1P = <A2C2P
= <B3C2P = <B3A3P
<PAB1 = <PC1B1 = <PC1A2 = <PB2A2
= <PB2C3 = <PA3C3.
That is, the two pairs into which AP divides <A have their equal counterparts at B1 and C1, again at C2 and B2, and finally both at A3. So ▲ABC and ▲A3 B3 C3 have equal angles at A and A3. Similarly, they have equal angles at B and B3. QED.