An+Unexpected+Right+Angle

Consider the following construction (see the figure below):
 * 1) Draw a square ABCD.
 * 2) Prolong AB, and choose E on this ray so that B is between A and E.
 * 3) Construct the line segments CE and DE.
 * 4) Let F be the point where DE intersects BC.
 * 5) Prolong AF, and let G be the point where the ray intersects CE.
 * 6) Construct the line segment BG.
 * 7) Let H be the point where DE intersects BG.

If you drag the point E, you will see that the line segments DE and BG appear to intersect at right angles. Can you prove that this is always the case?

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