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Essentially, two stars of identical mass but different temperatures are in a binary system. One planet is directly in their barycenter, tidally locked to one and therefore both stars. The effect I'm going for is one half of the planet is daylight by a bright white/blue star, and the other is "night" even though it's lit by the other star which is red.
Is this possible?
No, it's not a stable setup. You've got the planet at the two stars' L1 point, but that's dynamically unstable, and when it drifts away it will tend to drift further. Only the L4 and L5 points are dynamically stable, and so they're your only options for a planet that's linked to two stars in this way.
