# So many Circles

Geometry Level 5

$$\triangle A_0B_0C$$ has $$A_0B_0=13$$, $$B_0C=14$$, and $$CA_0=15$$.

A circle with diameter $$A_0B_0$$ is drawn; it intersects $$A_0C$$ at $$A_1$$ and $$B_0C$$ at $$B_1$$.

A circle with diameter $$A_1B_1$$ is drawn; it intersects $$A_1C$$ at $$A_2$$ and $$B_1C$$ at $$B_2$$.

This process continues infinitely. A circle with diameter $$A_kB_k$$ is drawn; it intersects $$A_kC$$ at $$A_{k+1}$$ and $$B_kC$$ at $$B_{k+1}$$.

The value of $\sum_{i=0}^{\infty} |A_iB_i|$ can be expressed as $$\dfrac{p}{q}$$ for positive coprime integers $$p,q$$. Find $$p+q$$.

Details and Assumptions

$$A_k\ne A_{k+1}$$ and $$B_k\ne B_{k+1}$$ for all integers $$k \ge 0$$.

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