Obviously AM GM

Geometry Level 3

Denote a variable \(a> 2\) such that \(A,B,C\) be three variable angles such that

\[ a \ \tan A + \sqrt{a^2+ 4 } \ \tan B + \sqrt{a^2 - 4} \ \tan C = 6a \]

What is the minimum value of \( \tan^2 A + \tan^2 B + \tan^2 C \)?

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