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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