Can we show through simple geometry rather than formulae or graphs that the best glide ratio occurs at the maximum ratio of Lift to Drag?
Let's start with a Lift/Drag/Weight diagram, and draw the velocity vector as well:

Since no thrust is involved, Lift+Weight+Drag must sum to zero. Let's shift the origin of these vectors around a bit:

Now, let's finish the velocity triangle, drawing a (horizontal distance), and b (vertical distance). The ratio between a and b is by definition the glide ratio.

The key idea here is to notice that the triangles L-W-D and V-a-b are similar, that is, they are the same triangle up to a scale.
The similitude goes: W &lt;-&gt; v, D &lt;-&gt; b, L &lt;-&gt; a.
This means that the ratio between a and b is the same as the ratio between L and D. The latter is of course the Lift-To-Drag ratio.
So, in any condition, the glide ratio is the same as the Lift-To-Drag ratio.
It then follows that the best glide ratio is achieved for the best Lift-To-Drag ratio.