A geometry problem by Neel Khare

Geometry Level 3

Let \(A,B\) and \(C\) be the angles of an acute-angled triangle. If the minimum value of \(\tan A \tan B \tan C \) can be expressed as \(m \sqrt n\), where \(m\) and \(n\) are positive integers with \(n\) square-free, find \(m^2+n^2-18\).

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