A triangle in the plane can be defined as the intersection of three half planes, each defined by the line passing through one of the segments and a normal direction pointing towards the interior of the triangle.
One thing is, if you flip the direction of all 3 normal directions (i.e. you take the complement of the 3 half spaces) then the intersection of those half spaces is the empty set, no matter the triangle.
I am having a hard time proving that postulate.
In fact, there is a simple reason which I am going to express in a graphical way.
Have a look at the following figure:
The three extended sides (Red, Green, Blue) of the triangle define 7 regions, with a 3 digits binary code for each one.
Let us explain how the first (red) digit is defined.
The regions bearing a left digit number $\color{red}{1}$ are those situated on the same side of the $\color{red}{\text{red}}$ line as the triangle, whereas it is a $\color{red}{0}$ for the others.
The other digits (the Green and the Blue ones) are attributed with similar rules.
Now the final touch. You may have noticed that among the $2^3=8$ theoretically possible codes, one is absent : code $\color{red}{0}\color{green}{0}\color{blue}{0}$.
What is this missing region, otherwise said, what is the meaning of this void region ? Precisely what you are looking for, the fact that no point can be situated simultaneously at the exterior of red, green and blue lines with respect to the triangle...