Prove that the visible lunar terminator (the boundary between the illuminated and dark parts of it) is a semi-ellipse

Prove that the visible lunar terminator (the boundary between the illuminated and dark parts of it) is a semi-ellipse, if we neglect the phenomena of parallax.

Since the moon is a ball, the true terminator is always a circle. In the projection for the observer, the circle turns, as is known, into an ellipse.

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