Spiraling Down

Calculus Level 5

Starting with the vertices P1=(0,1), P2=(1,1), P3=(1,0), P4=(0,0)P_1=(0,1),\ P_2=(1,1),\ P_3=(1,0),\ P_4=(0,0) of a square, we construct further points as follows:

  • P5P_5 is the midpoint of P1P2P_1P_2.
  • P6P_6 is the midpoint of P2P3P_2P_3.
  • P7P_7 is the midpoint of P3P4,P_3P_4, and so on.

The spiral path approaches a point P=(AC,BC)P_\infty=\left(\dfrac{A}{C},\dfrac{B}{C}\right) inside the square, where A,B,CA, B, C are coprime positive integers.

Find A+B+C.A+B+C.

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