system
Given the following post, respond as if you're commenting on Reddit:
human
Is there a lagrange point at a binary system's center of mass?
gpt
If the two masses were the same, it (the center of mass) would coincide with the L1 Lagrange point, but otherwise no. An object passing through the center of mass would have to orbit the heavier body faster (in degrees per second) than the lighter body, and so could not remain at the center of mass.

The center of mass is not (unless the two bodies are the same mass, as above) the point where gravity from the two bodies cancels out.