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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