Here's something that's puzzled me from time to time.

We all know that it is the shadow of the Earth on the Moon (or the lack of said shadow) that creates the phases of the moon that we see...such as half moon, quarter moon, new moon, full moon, etc...

Fine.

Now, here's the problem. I'm looking at a nice half moon today, easily visible in the bright blue afternoon sky on a brightly sunlit day. You can see the half of the moon that's lit by the sun, but the dark half is invisible. The moon has climbed about halfway to the Zenith.

The sun is still far above the western horizon, shining brightly.

Now here's my question. Given the fact that I can plainly see the entire moon, well above the Earth's horizon...and the entire sun, also well above the Earth's horizon, both with a lot of blue sky between them and the Earth's horizon....

How the hell can the Earth be interposing itself at this time between the sun and the moon in such a way as for the Earth's shadow to block out half the moon from the sunlight?

I can see plainly from where I am standing that that makes no geometrical sense at all, because the sun and the moon are both way "above" every part of the Earth that is visible from where I stand. WAY above it.

So how can any part of the Earth itself be getting in between them right now and causing its shadow to put half the moon in darkness?

This totally puzzles me. It appears to defy the normal rules of geometry. Light goes in a straight line, correct? If anything was in between the sun and the moon right now, it would have to be way up above me, I would think, since they are both way above the eastern and western horizons. Anyone got the explanation for this apparent anomaly?