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Is there a Lagrange point between the earth and the moon?

Yes. The Earth-Moon system has a Lagrange point L1, positioned between the Earth and the Moon, It is about 85% of the distance to the moon (about 320000km compared to 380000km.) A body at L1 would ...
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Is there a Lagrange point between the earth and the moon?

@JamesK's answer is good, but I'll address some of your points more specifically. Is there a Lagrange point between the Earth and Moon... Yes, it is called Earth-Moon L1 (to distinguish it from Sun-...
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Is there a Lagrange point between the earth and the moon?

Yes, the Earth-Moon system has Lagrange points, and Wikipedia has a short list of satellites that have (or will) orbited near L2, L4 or L5. Neither L1 nor L3 appear to be popular. The Moon's orbit ...
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