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Ellipse Proof: The sum of distances d(F_{1}, P) + d(F_{2}, P) remain constant as the point P moves along the curve.
- A line passing through P and the vertex S of the cone intersects the two circles at points P_{1} and P_{2}.
- As P moves along the curve, P_{1} and P_{2} move along the two circles.
- The distance from F_{1} to P is the same as the distance from P_{1} to P, because these distances are the lengths of the line segments PF_{1} and PP_{1}, which have a common endpoint P and are both tangent to the same sphere (G1) at their other endpoints.
- Likewise, by a symmetrical argument, the distance from F_{2} to P is the same as the distance from P_{2} to P.
- Consequently, the sum of distances d(F_{1}, P) + d(F_{2}, P) must be constant as P moves along the curve because the sum of distances d(P_{1}, P) + d(P_{2}, P) also remains constant. Therefore, the curve is an ellipse. This follows from the fact that P lies on the straight line from P_{1} to P_{2}, and the distance from P_{1} to P_{2} remains constant.
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