Critique of constant sum property

In the proof, we showed that all points on the cylindrical section have the constant sum property. Conversely, do all points in that same plane with that same constant sum lie on the cylinder—that is, on the ellipse?

We should think about how various properties of circles extend to ellipses. Some of the properties of circles are

• every radius to a point on a circle is perpendicular to the tangent line at that point;
  
  
• a circle with radius r centered at the origin has equation x ² + y ² = r ²;
  
  
• the circumference of a circle with radius r is 2¼r; and
  
  
• the area of a circle with radius r is ¼r ².

What are the analogies of these properties for ellipses in general?

Are there more patterns in the spheres we introduced to prove the constant sum property.