Why does the distances from the focus points of an ellipse to arbitrary point in the ellipse sum up to the length of the diameter of the ellipse in the major axis? In other words, how to prove:
I have seen videos and papers, where this is proved by using the picture on the left. But does this hold in every case no matter where the blue dot $(x,y)$ is on the ellipse?
Question: How to prove that $d_1+d_2 = \hat{d}_1+\hat{d}_2= 2a$ always?
Thnx for any help =)
One way to characterize an ellipse is by stating that this is the set of all points which have equal sum of distances to the pair of foci. So if one uses this as the definition, then if $d_1+d_2=2a$ holds in the left picture, then it must hold for all $\hat d_1+\hat d_2$.
If you use a different definition of an ellipse, you might have to prove the constant distance sum up front, but this depends very much on which definition you use instead. The way I understand the OP, this is not the case here.