Nasty Triangles 3

Geometry Level 5

Given a circle and \(\triangle ABC\), the circle passes through \(A\) and \(B\) and intersects \(\triangle ABC\) at point \(D\) and \(E\). Extend \(AB\) and \(DE\) so that they intersect at point \(F\), connect line \(CF\).

If \(AB=3\), \(AD=1\), \(DC=2\), and \(\frac{AE}{BD}=\frac{\sqrt5}{2}\), find the length of \(CF\).


Also try Nasty Triangles 4.

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