Dandelin spheres -Conic Sections Proof of Circle Ellipse Parabola Hyperbola

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• Wikipedia: Dandelin spheres

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.