The Three Point Conjecture

Don't get too excited about the title. It's simply a puzzle(The three utilities puzzle).

There are three houses and three factories(water,gas,electricity) in a city block.Each house should be supplied with water,gas and electricity.The supply lines should not overlap each other.Draw the connections.

In short there are 2 sets of three points.Each point in Set-1 should be connected to every point in Set-2 without the lines overlapping.

Seems easy enough.But there's a catch, no matter how you try, you won't be able to make one last connection.You don't want to know how much time I spend on it and was convinced of its impossibility.

Now one last challenge remained - the proof.
The difficult part about the proof was that I've got no idea where to begin or which path to follow.Proving something like this was entirely new. It disturbed me day and night. It's amazing how desperation could make you do things that you thought you couldn't. And this fine morning something strange struck me.

Proof:

Let the red points represent the factories and blue be the houses

Now lets connect 2 red dots to all the blue dots.

Now lets displace the points a bit.There is no loss of generality in doing so.Such a configuration divide the whole plane into 3 areas. The red dot at the bottom can be brought back to the top, elongating the 3 lines connected to it.

the plane will be divided into 3 areas however curved the lines may be.
(Because:There are 3 curved lines starting at red going through blue and ending on red.These 3 curved lines have the same end points without overlaps.Thus they divide the plane into 3 distinct areas.)

Now place the remaining red point in any of the 3 enclosed areas.The third point can only connect 2 of the 3 blue points.
Thus proved...