As said before, gravity of moon and earth will cancel each other at the five Lagrangian points. For further information you could read the article at Wikipedia: http://en.wikipedia.org/wiki/Lagrangian_point
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Proving that there are five Lagrangian points and determining their locations is left as a homework problem.
There are FIVE points in the earth moon system where gravity "cancels" - they are called the Lagrangian points.