Day 18: An Incredible Result

Geometry Level 4

A triangle \(ABC\) has a right angle at \(B\). A semicircle is inscribed within on the side \(BC\) and its diameter is equal to \(AB\).

If the ratio \(\frac{AB}{BC}\) can be written in the form \(\frac{a}{b}\) where \(a\) and \(b\) are positive coprime integers, find \(a+b\).

This problem is part of the set Advent Calendar 2014.

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