I am struggling with understanding the formula for the area of a regular hexagon. In all resources that I have referred to, it seems to be taken for granted that a regular hexagon consists of six equilateral triangles. But no explanations are provided on how we can be sure about that. Not only do I not understand how we can be 100 % sure that those triangles are equilateral, but I am also not quite sure about the certainty of the claim that all those inner lines (diagonals inside the polygon) will intersect in one point.
Is there any simple way to eliminate all those doubts?
Consider regular hexagon $ABCDEF$. Let $O$ be the centre of the circumscribed circle of $ABC$. The rotation around $O$ that maps $A\mapsto B$ must map $B\mapsto C$ and maps $C$ to the point $X$ such that $CX=BC$ and $\angle XCB=\angle CBA$, in other words, we have $X=D$, $C\mapsto D$. By the same argument, our rotation maps $D\mapsto E$, $E\mapsto F$ and $F\mapsto A$. It follows that $OAB$, $OBC$, etc. are isosceles and as the angles at $O$ must add up to a full angle, all six central angles must be $60^\circ$.