Charlie, don't lose heart. I don't know why all of the fuss. I didn't think twice about writing style when I read your post. You're question was legible, which is all I ask.

Back to the point, I'm a P.E. and I've never heard of a balancing beam. The photos aren't enough to tell what's going on, but it looks like you have two beams at 90 degrees to each other, over a steel post. This is usually done when the floor framing that the beams support spans in different directions and/or there's a change in floor elevations. My only comments, based on what I can tell from the photos, are that there are no web stiffeners (not necessarily a problem; residential loads are usually relatively light) and it doesn't appear that there are any mechanical connections (beam to beam or beam to post).

As a home inspector, you should probably just note that you have some questions and possibly recommend a P.E. to evaluate. This might be overkill, though, if the house is older and there are no signs of movement (wall cracks, etc.).