# 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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