$\overrightarrow{p}=\left(k\tan A-\sqrt{k^2+9}\tan B\right)\overrightarrow{i}+\left(\sqrt{k^2+9}\tan C-\sqrt{k^2-9}\tan A\right)\overrightarrow{j}+\left(\sqrt{k^2-9}\tan B-k\tan C\right)\overrightarrow{k}$ 

and 

$\left|\overrightarrow{P}\right|=9k$ . 

Then the least value of ${\tan }^2 A+{\tan }^2 B+{\tan }^2 C=$

$\overrightarrow{P}=\left(\sqrt{k^2+9}\overrightarrow{i}+k\overrightarrow{j}+\sqrt{k^2-9}\overrightarrow{k}\right)\times (\tan C\overrightarrow{i}+\tan B\overrightarrow{j}+\tan A\overrightarrow{k})$$\left|\overrightarrow{P}\right|=\sqrt{k^2+9+k^2+k^2-9}\sqrt{{\tan }^2 C+{\tan }^2 B+{\tan }^2 A}\sin \theta =9k$$\sqrt{3}K\sqrt{{\tan }^2 C+{\tan }^2 B+{\tan }^2 A}\ge \left(9k\right)$

${\tan }^2 A+{\tan }^2 B+{\tan }^2 C\ge 27$