INTRODUCING 5 - days-a-week problem solving session for Math Olympiad and ISI Entrance. Learn More 

December 16, 2018

Spiral Similarity of cyclic quadrilaterals

ABC be any triangle. P is any point inside the triangle ABC. ( PA_1, PB_1, PC_1 ) be the perpendiculars dropped from P on the sides BC, CA and AB respectively. ( A_1 B_1 C_1) constitutes a pedal triangle.

Also see

Advanced Math Olympiad Program

Drop perpendiculars from P on ( A_1 B_1, , B_1 C_1, C_1 A_1 ) at ( C_2, A_2, B_2 ) respectively. ( A_2 B_2 C_2 ) known as the second pedal triangle.

Finally, repeat the process to have the third pedal triangle (A_3 B_3 C_3 ).

Proposition (easy angle chasing): The third pedal triangle is similar to the original triangle ( ( \Delta ABC \sim \Delta A_3 B_3 C_3 ) )

Spiral Similarity Cyclic Quadrilateral Pedal triangle

Spiral Similarity

Notice that quadrilateral ( q_1 = P A_1 B C_1 ) is cyclic (why?). Rotate ( q_1 ) by ( 180^\circ ) and dilate it by a factor of ( \frac {1}{8} ). This spiral similarity sends the vertex B to ( B_3 ).

Exercise 1: Proof this using complex bashing or otherwise.

Exercise 2: Normalize by recreating the process in an equilateral triangle.

Remark: It is interesting to note that ( P A_1 B C_1 ) appears to be spirally similar to ( PB_2 C_1 A_2 ) and ( PC_3 A_2 B_3 ) but that does not happen.

 

Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com