<s>[INST] How do I find the diameters of the circles in this geometry puzzle? My family and I like to do a daily quiz but this particular question has had us baffled for weeks. Please help. We only have basic mathematical knowledge.
 
 [/INST] $\triangle DCE$, $\triangle ECA$ and $\triangle AED$ are right-angled triangles. We therefore have
$$ \begin{align}
    AD^2 &= AE^2 + DE^2 \\
         &= AC^2 + CE^2 + EC^2 + CD^2 \\
         &= AC^2 + (AC - 5)^2 + (AC - 5)^2 + (AC - 9)^2
\end{align}$$
But we also have $AD = 2AC - 9$, so we can solve for $AD$:
$$ \begin{align}
    (2AC - 9)^2 &= AC^2 + (AC - 5)^2 + (AC - 5)^2 + (AC - 9)^2 \\
    4AC^2 - 36 AC + 81 &= AC^2 + 2AC^2 - 20AC + 50 + AC^2 - 18AC + 81 \\
    2AC &= 50 \\
    AD &= 41
\end{align}$$
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