Escape the Plane

One Sunday morning, you awake to find yourself completely alone on an infinite, flat plane. You don’t remember much about the night before, other than that you may have pissed off a wizard. Next to you, you find a palette with countably infinite colors, and a note, commanding you thus:

You must paint every point on this plane, such that I will never be able to find a triangle with vertices of the same color and rational area.

If you can manage this task, the wizard will let you go free – fail, and you’re trapped forever. You don’t doubt the wizard’s abilities, so no cheap tricks here. Considering the problem, you get to work – and an uncountably infinite amount of time later, the wizard stands beside you, admiring your handiwork.

Does the wizard set you free?

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EDIT: To eliminate lateral thinking answers based on the framing of the question, here’s a formal mathematical statement of the puzzle:

Does there exist a coloring of $mathbb{R}^2$ such that it is impossible to find a triangle with vertices of the same color and rational area?