Can you determine how fast you were going at a midpoint of a trip by calculating average speed from the entire trip?

You can't. The average doesn't tell you anything about any single point during the trip.

If I start out this morning and arrive somewhere 1,000 miles away tomorrow afternoon,

my average speed for the entire trip could be 35 miles per hour. But at the mid-point

of the trip ... either the mid-miles or the mid-time ... I could very well have been in a

motel asleep. You can't tell.