One way to define an ellipse is as the set of all points whose sum of distances from two foci (the two blue non-moving points above) is constant. Since each of the moving dots travels from one focus to the ellipse then to the other focus, each dot travels the same distance. All dots move at the same speed, traveling the same distance, meaning they arrive at the other focus at the same time. (This has an interesting application to the study of the reflective property of parabolas, shown in a different post.)

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