Let us suppose, as on the figure, that I have a (given) hexagon $DEFIHG$. Then, I build the hexagon $KLMNOP$, which is homothetic of ratio $\alpha$ of the first one ($\alpha$ is a parameter). That is, the outer hexagon is nothing more than the inner "scaled" by a factor $\alpha$.
Remark that each hexagon has three couples of sides parallel, as suggested on the figure. It is an assumption.
I am looking for a criterion such as $\alpha \geq ...$ for which there exists a triangle enclosed between the two hexagons, as the green triangle. By this, I mean, a triangle whose vertices are inside the outer hexagon and the edges do not cross the inner hexagon.
I would also be interested by the conditions leading to two different triangles
Thanks
Say you have a hexagon $ABCDEF$ (as in diagram below) and want to construct a hexagon homothetic to it, containing triangle $HIJ$, which is obtained by producing sides $BC$, $DE$, $FA$.
The smaller of such hexagons will have three sides passing through $H$, $I$, $J$ and parallel to $EF$, $CD$, $AB$ respectively, lying then on triangle $NOP$ in diagram below. But triangle $NOP$ is homothetic to triangle $KLM$, obtained by producing sides $AB$, $CD$, $EF$ of the original hexagon. The center of homothety is $Q$, intersection of lines $KN$, $LO$ and $MP$ and the homothety ratio is $$ \alpha={QN\over QK}={QO\over QL}={QP\over QM}. $$
This homothety carries hexagon $ABCDEF$ to the requested minimal hexagon.