Sum of squared tangents in a tetrahedron section

High school · ●●●●○ 100 pts

A regular tetrahedron is intersected by a plane that is perpendicular to one of its faces and meets that face along a line

e

. The intersection lines of this plane with the other three face planes of the tetrahedron form angles

\varphi_1, \varphi_2, \varphi_3

with

e

. Determine the value of

\tan^2 \varphi_1+\tan^2 \varphi_2+\tan^2 \varphi_3