The foci of an ellipse, reflected across its tangents

James Tanton [url=http://twitter.com/jamestanton/status/608240289136836608]asked[/url]: Reflect an [color=#ff0000][b]ellipse[/b][/color] across each [color=#0000ff][b]tangent line[/b][/color] to it. What curve(s) do the images of its two foci trace? The [color=#ff0000][b]ellipse[/b][/color] shown below can be modified by dragging the three [color=#ff0000][b]red points[/b][/color]. The [color=#0000ff][b]tangent line[/b][/color] can be chosen by dragging the [color=#0000ff][b]blue point[/b][/color]. The reflections of the foci, and the curves they trace, are shown in [color=#0a971e]green[/color] and [color=#b20ea8]purple[/color].

Are the generated curves really circles, as they appear to be? If so, how can this be proved?