Easy, stand on

Taking this in one dimension, by the given heuristic, where is your distance from girls one and two. By Geo 3 the locus of points is actually the Apollonius Circle but let's restrict this to the line segment connecting the two girls.

If you stand somewhere in the range we can replace with respectively, so we have which arranges to

Not surprisingly, we also know this is the answer of using basic physics or center of mass.

This is where things get tricky. We would expect the center of mass points idea to generalize, that is, stand on but consider the case when We shouldn't expect to closer to one girl than the other, so we stand at a point that's equidistant to each girl, so it's expected we should stand on the circumcenter.

How did this happen!?!?

So I thought about this for a while. It appears the morally correct location to stand is the circumcenter, but clearly this doesn't generalize for all The idea is that the locus of points we're allowed to stand on given is a perpendicular line going through the center of mass of girls one and two. But clearly these three perpendicular lines don't have to intersect.

In the case of if this will only work if the polygon is cyclic.

The center of mass formula, though, generalizes to all , even infinite which we handle with a double integral.

It seems the engineer's answer would be the center of mass, as it seems to make the most sense while being computable every time.

It also backed by the derivation of torque in physics.

The biggest problem it seems with the first approach is that by focusing on two girls at a time, the resulting loci don't intersect. So perhaps we can define some sort of error function that tracks how badly we fail, so to "compromise" would be to minimize this error function. Perhaps we can prove this error function is minimal when we arrive at the center of mass.

A perhaps analogous problem is to consider the boy as a proton and the girls as varying electric charges. The problem here is that each charge creates an electric field that affects all the other charges. So suppose we can fix them all in place somehow. However, in this case electric force = electric field x electric charge replaces the formula mass = density x volume when carrying out the calculation.

Perhaps the party boy can look at the movement of the proton for some inspiration?