how do you get a line to go to three houses from three circles without each of them touching or crossing each?

Answer 1

This is impossible unless you use some trickery.

Basically, you connect as many as you can directly (see hints below), but you'll find that there's one connection you can't make. (One of the houses is connected to two circles, but is not connected to the third circle.)

At that point, you take one of the houses that is already connected to that particular circle, and run a connection THROUGH that house to the house that has not been connected.

That probably is not legitimate, but in puzzle books that's the way the problem is always solved.

Hintes for making the initial connections:
Start with the middle circle and connect it directly to each house.
Then take the left circle and connect it directly to the left house, but connect it to the other two houses by drawing lines around the left side of the row of houses, and connect the houses from the rear.
Finally, go to the right circle. You can connect it directly to the right house. and you can go around the middle circle to connect to the left house. But you can't get to the middle house.
Now you take the connection you've made from the right circle to the right house, and extend it from the right house to the middle house. Then you have all the connections, and the only question is whether that last connection was legitimate.

Answer 2

I believe you are asking the question wrong. If I am correct, you want to draw three lines from each of three circles (9 total lines) to three houses, so that each circle is connected to each house exactly once by a line. The solution: impossible.
This is an old problem in an area of mathematics called Graph Theory. It is often referred to as the utilities problem: connect a gas line, water line, and electricity line to each of three houses, each house having it's own direct connection. I cannot remember the proof of the problem, but do know that it cannot be done without at least one crossing.
A similar problem was the Bridges of Konigsburg problem. The residents of a town (Konigsburg) enjoyed walking around town after church each week. The town had a river running through it, with two islands in the middle. There were also seven bridges in town (one connected the islands, two from each bank connected to one island, and one from each bank connected to the second island). The residents wanted to start at one bridge, cross all the others exactly once, and have returned to their starting point. Euler proved this also was impossible.