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? For an unpowered glider, can we show through simple geometry involving force vector diagrams, rather than mathematical formulae or graphs, that the best still-air glide ratio is achieved at the angle-of-attack that gives 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.