Question 65
The parallelogram $ABCD$ has center $O$, $AD = DB$ and $BA \gt BC$. The point $V$ is outside the plane $(ABC)$ such that $VD \perp (ABC)$. Point $T$ is the foot of the perpendicular from $D$ onto $AB$ and $E, F, G$ are the feet of the bisectors of the angles $\angle VDA, \angle VDB, \angle VDC$ respectively (where $E \in VA, F \in VB, G \in CV$). Prove that:
i) $EF \parallel (ABC);$
ii) $m(\angle((VAB), (ABC))) = 45^\circ$ if and only if $VD = DT;$
iii) if $VD = DT$, then $\tan(\angle((GEF), (ABC))) = \frac{1}{2}$ if and only if $\frac{BA}{BC} = \frac{6}{5}.$
i) $EF \parallel (ABC);$
ii) $m(\angle((VAB), (ABC))) = 45^\circ$ if and only if $VD = DT;$
iii) if $VD = DT$, then $\tan(\angle((GEF), (ABC))) = \frac{1}{2}$ if and only if $\frac{BA}{BC} = \frac{6}{5}.$