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The foci of an ellipse, reflected across its tangents (2)

James Tanton asked: Reflect an ellipse across each tangent line to it. What curve(s) do the images of its two foci trace? The ellipse shown below can be modified by dragging the three red points.
Perhaps surprisingly, the curves generated by the reflected foci are circles. As pointed out by Matt Enlow, this is because, in the visual above, the segment lengths a and b must always sum to the same constant.