In $\triangle ABC$,if $\tan A + \tan B + \tan C = 6$ and $\tan A \cdot \tan B = 2$,then $\tan C = \dots$

In a $\triangle ABC$,if $\tan A : \tan B : \tan C = 1 : 2 : 3$ and $\sin A : \sin B : \sin C = \sqrt{5} : 2\sqrt{2} : k$,then $k =$

If $P$ is a point on the altitude $AD$ of the $\triangle ABC$,and $\angle ABP = \frac{2B}{3}$,then $AP$ is equal to

In $\triangle ABC$, right-angled at $A$, the circumradius, inradius, and radius of the excircle opposite to $A$ are respectively in the ratio $2:5:\lambda$. Then the roots of the equation $x^2-(\lambda-5)x+(\lambda-6)=0$ are:

If $\alpha, \beta$ and $\gamma$ are the lengths of the altitudes of a $\triangle ABC$ with area $\Delta$,then $\frac{\Delta^2}{R^2}\left(\frac{1}{\alpha^2}+\frac{1}{\beta^2}+\frac{1}{\gamma^2}\right)$ is equal to

In $\Delta ABC$,the expression $a(\cos^2 B + \cos^2 C) + \cos A(c \cos C + b \cos B)$ is equal to: