I don't think the note "Since the triangle got rotated halfway around the circle, the tip must end up exactly opposite from where it started. That's why the dia

I don't think the note "Since the triangle got rotated halfway around the circle, the tip must end up exactly opposite from where it started. That's why the diagonal of the box is a diameter." is needed, nor do you need to show the diagonals of the parallelogram are equally long. Rather, it suffices to simply state, the only parellelogram that is inscribed in a circle is a rectangle. For the less rigorous, that statement is obviously true. For complete rigor, it is sufficient to argue that the center of the circle, combined with the vertices of the parallelogram forms an isoceles triangle. So the centre of the circle must lie on the bisector all edges of a parallelogram. But on a non-rectangle parellelogram the bisectors of opposite edges never intersect.