Very difficult geometry - challenge!

[mathnerd3141]

Consider a pentagon P. Select each combination of three points, and connect the incenters of the triangles determined by these combinations into a polygon A with maximized area. Extend each of the sides into a line, and determine the circle tangent to each possible combination of three lines. Connect the centers of all of these circles into a polygon B with maximized area. Find the minimum possible perimeter of P if the area of the intersection of A and B is equal to 10.
Or prove that this problem isn't possible.

It's not my homework; I just made it up myself. I can't get anywhere at all on it.

Any ideas?

(Also, I'm not certain it fits under this category "Geometry and Trig.")

-mathnerd3141

 

[mathnerd3141]

Regular pentagon?

10 combos of "three points"; right? Like, if labelled A,B,C,D,E:
1: ABC
2: ABD
...
10:CDE

Still hard to "see" what you're trying to do;
can you show an example...like a diagram or something?

You're calling 'em polygon A and polygon B: if as you say you've "done the work",
why do you not have the number of sides for these?

Finally, what were you smoking when you made that up?

Respectively:

Not necessarily.

Yes. (5 choose 3 = 5!/(3!2!) = 10)

I don't exactly have something that can make such a diagram. (Paint just doesn't work in this kind of situation, and I've tried a few utilities, with limited success.)

The number of sides is not necessarily 10. Sometimes the centers coincide or are (three in a row) collinear.

I'm just mathematically inclined - I'm 13 .