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equilateral triangle inscribed in an ellipse

A high-schooler I know was given the following problem:

> In the ellipse $x^2+3y^2=12$ is inscribed an equilateral triangle. One of the triangle's vertices is at the point $(0,-2)$. Find the triangle's other vertices.

The book has one answer: $(\pm1.2\sqrt3,1.6)$. But I know of two more answers: $(0,2)$ and $(\pm\sqrt{12},0)$. Are there any more?