EUR 10.-family housework monitoring system, who has to do the dishes next?

Dear Starminds,

here i forward a question from my office mate, which is mathematically quite nice.

The starting point is his system to monitor the equal effort done in the housework between him and his wife. They have a measure with their names at both ends. Every time someone does a task, he/she moves a pin one unit step towards his/her name. The goal is to keep the family harmonic by keeping the pin more or less in the middle between the two names. 

Currently their son is too young to be introduced to cleaning the bathrooms, but there will come a day, when he has to be incorporated into that system. And maybe, he will get siblings...

So, here comes the question:

How has a three or more member solution of the above monitioring system to look like?

Mathematically, this means: is there a way to build a manifold in 2D, embedded in 3D obviously, that keeps the sum of distances to three particular points fixed? How does it look like? How is the pin moved on that manifold?

Not necessary for the full rating of the solution are the additional tasks: How about more than three points? Is there a way to incorporate a weighting such that the little one has only to do a fraction of his parents housework? More explicitly: Is it possible to design the manifold such that a unit step towards one name draws away the pin from the other names in a nonequal manner?

have fun!

   Hecke