Problem to shake your mind

Geometry Level 5

If \(a\) is a real constant and \( A,B \) and \(C\) are variable angles. They are related to the constraint \( \sqrt{a^{2}-13}\tan{A} + a\tan{B} + \sqrt {a^{2}+13}\tan{C} = 6a\) . Then find minimum positive value of \(\tan^{2}{A} + \tan^{2}{B} + \tan^{2}{C}\).


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