I am trying to learn how to derive the equation of an ellipse, from this website (https://people.richland.edu/james/lecture/m116/conics/elldef.html). I am struggling, however, to prove to myself why d1 + d2 = 2a (see the diagram).
I get that when you stretch the "rope" against the major axis, you will see that d1 is equal to the length -c-(-a). But I'm struggling to prove that the same will occur when you stretch the rope on the other side of the ellipse, the side with positive c and a values. I get that by definition, the ellipse has to be symmetrical. But I want to prove that how constructing the ellipse using the "rope" would force the ellipse to be symmetrical. Can someone please show me how to do this?