(Solution) - Let C be the curve obtained by intersecting a cylinder -(2025 Original AI-Free Solution)

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Let C be the curve obtained by intersecting a cylinder of radius r and a plane. Insert two spheres of radius r into the cylinder above and below the plane, and let F1 and F2 be the points where the plane is tangent to the spheres [Figure 16(A)]. Let K be the vertical distance between the equators of the two spheres. Rediscover Archimedes's proof that C is an ellipse by showing that every point P on C satisfies
PF1 + PF2 = K