Another way of looking at push pull is from the same foot. We swing the LF back by first pulling it with the ball of the (RF) and then pushing it with the heal of (RF). NOTE: I still have to read the previous replys Thanks Dave.

No,you push from the departing foot and then allow your weight to coast to arrival? To arrive on the ball of foot which then has to pull the weight onto the foot.
Using the arriving leg to redirect your path ? Yes by pushing of the heal of that foot.
You slow your arrival? Yes you slow your arrival by the speed at which you pass over the whole foot"ball & heal"

If we agree you slow your arrival and re-direct it, why do you persist in this idea of pulling on the arriving foot? Pulling makes you go faster after all. And you are already going plenty fast enough if you use the departing leg properly.

When we place our weight on and over the ball of the back foot we then have to pull it onto the heal,but also for the same reason that Geoffrey Hearn says we pull our weight over and through the front foot. I you can't agree that Geoffry Hearn is right on the forward step then you won't agree that it happens on the backword step. Soum, I am not disagreing with you,just stating another point of view. I don,t have the knowledge to give correct answers.

"When we place our weight on and over the ball of the back foot we then have to pull it onto the heal"

Only if you insist on outstepping your body flight. Since this forces you to break contact with your partner or arch you back to make room for the moving leg, there's two very good reasons (efficiency and appearance) to keep your leg swing in natural proportion to your body flight, such that arrival over the foot is automatic with no additional work required.

In practice, dances who have to pull themselves onto their feet cannot be considered to have body flight at all. They tend to be those who let the body fully or nearly stop over each step, reach the foot out, and then labouriously transfer from foot to foot. That's closer to tango, NOT waltz, foxtrot, quickstep or viennese.

At the end of the third step of the foxtrot reverse turn a top pro on vidio has his Rf well back the right knee close to the floor his weight is well forward of the foot ,the only way he can get his weight onto that fourth step(RF) is to pull it on. You can also see his left heal sliding along the floor for some distance before that left foot swings to the side for the fifth step. Pictures speak louder than words.

I must confess - i wish i could see the pictures because i've never seen what you describe. Which video is it? - i'll try to get my hands on it. I have Sinkinson and Hilton teaching videos and i've never seen anything like this on them.
When you say reverse turn i assume you mean with the lady doing a heel turn. Isn't the first step a slow on the man's left, - second step a quick on man's right, having made most of the turn, - the third step man's left foot back..etc. All these steps are made with a pushing action from the man's standing leg (and to some extent, controlled use of body momentum). Even sprinters don't pull themselves onto the (front) foot.

"the only way he can get his weight onto that fourth step(RF) is to pull it on."

No, he gets his weight to the new foot from executing a powerful pushing action as he releases the toe of his departing foot - or at least he would if he wasn't too original to use the classic technique. But he wouldn't need to push all that much anyway, because there should be substantial body flight carrying him onto the foot. Pushing only extends the possible distance.

Incidentally, you've posed a false premise. The end of the third step is defined as when his feet are passing, so it's impossible for him to have one foot back at that point. What you are actually talking about is the midpoint of the fourth step.

I have read all the above and can't see what the the argument is'. If I do not push off the back foot going forward, or do not push off the rear foot when going backwards then I am standing still
going nowhere.