Ellipse Proof: The sum of distances d(F1, P) + d(F2, 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 P1 and P2.
- As P moves along the curve, P1 and P2 move along the two circles.
- The distance from F1 to P is the same as the distance from P1 to P, because these distances are the lengths of the line segments PF1 and PP1, 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 F2 to P is the same as the distance from P2 to P.
- Consequently, the sum of distances d(F1, P) + d(F2, P) must be constant as P moves along the curve because the sum of distances d(P1, P) + d(P2, P) also remains constant. Therefore, the curve is an ellipse. This follows from the fact that P lies on the straight line from P1 to P2, and the distance from P1 to P2 remains constant.