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Anonymous at Sun, 9 Jun 2024 13:31:32 UTC No. 16224374

Consider a point F and a line l.

Let the point P be the foot of the perpendicular to l passing through F.

Let E be the ellipse with focus F, directrix l and eccentricity r.

Then E intersects the line FP at two points A and B. Assuming AP > BP, then AP/FP = 1/(1 - r), which, as you may notice, is awfully reminiscent of the formula for the sum of an infinite geometric series. Is this a coincidence?