The coordinates of two consecutive vertices $A$ and $B$ of a regular hexagon $ABCDEF$ are $(1, 0)$ and $(2, 0)$, respectively. The equation of the diagonal $CE$ is

Refer to the figure attached. $ABCDEF$ is a regular hexagon with side length $1$ with other sides above $x-$ axis. The coordinates of $A$ and $B$ are given.

Length of $AE$ can be found using sine rule in triangle $FEA$

$\dfrac{\sin {120^{o}}}{AE}=\dfrac{\sin {30^{o}}}{1}$

$\Rightarrow AE = \sqrt{3}$

Since for the given hexagon $AE \perp AB$, we get the coordinates of $E$ as $(1, \sqrt{3})$....(1)

Now, $FC || AB$ and $\angle {ECF} = 30^{o}$

Hence, slope of $CE$ is $\tan {120^{o}}$....(2)

From (1) and (2) equation of $CE$ is $y- \sqrt 3 = \tan {120^{o}}(x-1)$

Simplifying we get

$\sqrt{3}y + x =4$