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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.

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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.

 

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