Sequences in Geometry

Geometry Level 5

Triangle \(ABC\) is such that \(AB = 3\), \(BC = 4\), and \(CA = 5\). For all integers \(n \geq 1\),

  • Let \(P_1\) be some point on \(CA\) between \(C\) and \(A\) such that \(BP_1\) is not an altitude.
  • Let \(l_n\) be the line through \(B\) and \(P_n\).
  • Let \(G_n\) be the circle with center \(C\) tangent to \(l_n\).
  • Let \(r_n\) be the radius of \(G_n\).
  • Let \(P_{n+1}\) be the intersection of \(G_n\) with \(CA\) between \(C\) and \(A\).

These sequences of geometric figures can be seen in the diagram above, with the labels removed. (The diagram is finite, but the sequences are infinite.)

The sequence of real numbers \(\{r_i\}\), for all integers \(i \geq 1\), satisfies this recurrence relation:

\[r_{i + 1} = \frac{ar_i}{\sqrt{br_i^2 - cr_i + d}},\]

for some positive integers \(a, b, c, d\). Find \(a + b + c + d\).


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