G4.4 In Figure $2, \triangle A B C$ is an equilateral triangle, points $M$ and $N$ are the midpoints of sides $A B$ and $A C$ respectively, and $F$ is the intersection of the line $M N$ with the circle $A B C$.
If $d=\frac{M F}{M N}$, find the value of $d$.
Let $O$ be the centre, join $A O$.
Suppose $M N$ intersects $A O$ at $H$.
Produce $F N M$ to meet the circle at $E$.
Then it is easy to show that:
$M N / / B C$ (mid-point theorem)
$\triangle A M O \cong \triangle A N O$ (SSS)
$\triangle A M H \cong \triangle A N H$ (SAS)
$A O \perp M N$ and $M H=H N(\cong \Delta ' s)$
$E H=H F(\perp$ from centre bisect chords $)$
Let $E M=t, M N=a, N F=p$.
$t=E H-M H=H F-H N=p$
By intersecting chords theorem,
$$
\begin{array}{l}
A N \times N C=F N \times N E \\
a^{2}=p(p+a) \\
p^{2}+a p-a^{2}=0 \\
\left(\frac{p}{a}\right)^{2}+\frac{p}{a}-1=0 \\
\frac{p}{a}=\frac{-1+\sqrt{5}}{2} \text { or } \frac{-1-\sqrt{5}}{2} \\
d=\frac{M F}{M N}=\frac{a+p}{a} \\
=1+\frac{-1+\sqrt{5}}{2} \\
=\frac{1+\sqrt{5}}{2}
\end{array}
$$