Wondering about Last Night's Moon.

I went outside with the dogs the other night and looked up at the moon.  The dogs went back in and I was still out there looking up at the moon.  In my simple mind I couldn't figure out exactly why the moon looked like it did.  It looked to me like it was exactly a half moon.  Yes, I know that unless the earth eclipses the moon that the sun hits half the moon all the time just as it does the earth, but that we are not always in the right position to see it that way.

So, why would I be wondering about it?  Well, what I couldn't figure out was if the moon was exactly half illuminated by the sun, the sun had to be directly west of it, since it was the west side of the moon that was bright.  But, it was going on 23:00 and the sun had set a long time before.  Now, if you don't believe me, draw a picture.  Draw a big round circle to represent the earth, a smaller one directly above the big one to represent the moon.  Now draw a line vertically through the moon and blacken in the left half.  Then directly way off to the right of the moon, draw another circle to represent the sun.  This sketch would represent what I believed I saw last night.

Now here is the problem as I see it.  Draw a line from the center of the sun to the center of the earth.  Then draw a line perpendicular to that line through the center of the earth.  Darken in the side of the earth away from the sun.  Now, if you were at the top of the earth looking straight up at the moon so that your would see the moon half illuminated by the sun, you would be standing in the illuminated part of the earth!!  Yes, but it was eleven pm last night and it was dark. . . take my word for it, except for the moonlight.

 

The above sketch I make shows what I am talking about.  Although it isn't to scale, it does show that if I really could see a half illuminated moon, it should have been closer to sunset time then at an hour before midnight.  I must have missed something in my assumptions, because it was dark for sure.  Maybe you all can explain it to me.  Now you all go out at night and look up, you may be surprised what you see.  Mars is very bright and will get to its brightest for a while next month.  Have a great day now, you hear?