Given any triangle, we can build three parabolas, each with focus on one vertex and with directrix the opposing side, as illustrated here:
My first conjecture, likely trivial, is that, given any triangle,
The three parabolas never intersect, but they are tangent to one another in at most three points.
For instance, in case of an equilateral triangle, it seems that the three parabolas "touch" each other in three points $E,F,G$
However, it is not obvious to me whether the equilateral triangle is the only case in which we can find three tangential points $E,F,G$.
The question is, then:
In which conditions (on the initial triangle) can we find three, two, one or no tangential points?
I apologize in case the question is trivial. But thank you very much for your hints, comments, suggestions!