# Fibonacci Triangles

Geometry Level 5

Let $$F_n$$ be the $$n$$th number in the Fibonacci Sequence. Consider the 3 points $$(F_{30},F_{31}), (F_{32},F_{33}), (F_{34},F_{35})$$ in the Cartesian plane. You are allowed to repeatedly apply the following operation:

Let $$P$$ be any one of the three points in the plane and let $$Q,R$$ be the other points. Draw the perpendicular bisector of $$QR$$ and let $$S$$ be the closest point on this line to $$P$$. Move $$P$$ to any point $$T$$ in the plane such that $$S$$ is also the closest point on the perpendicular bisector of $$QR$$ to $$T$$.

After some sequence of operations, two of the points end up at $$(0,0)$$ and $$(100,0)$$. If the third point is in the first quadrant, the largest possible value of its $$y$$-coordinate can be expressed as $$\frac{a}{b}$$ where $$a$$ and $$b$$ are coprime positive integers. What is the value of $$a + b$$?

Details and assumptions

The Fibonacci sequence is defined by $$F_1 = 1, F_2 = 1$$ and $$F_{n+2} = F_{n+1} + F_{n}$$ for $$n \geq 1$$.

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