While studying ellipses, I've read the following proof which I can't understand.
Taking two point $F_1$ and $F_2$,we can define an ellipse as the set of points $Z$ sucht that $ZF_1+ZF_2$ has some constant value. We call $F_1$ and $F_2$ the foci of the ellipse. To see that this new definition of an ellipse satisfies our equation form,we apply the distance formula. First we draw the axes of the ellipse and note that $CD=2b$ and $AB=2a$. Now we can find the constant sum $ZF_1+ZF_2$.
Letting $Z$ be $A$ we have $ZF_1+ZF_2=AF_1 +AF_2=BF_2+AF_2=AB=2a$ where we note that $AF_1=BF_2$.
Now what is unclear to me is whether $AF_1=BF_2$ is true because of the fact that we have $ZF_1 + ZF_2$ constant or because the author has a priori defined $F_1$ and $F_2$ as equidistant.
If the first option is the true one,can you help me understanding the why ?
For all points $Z$, we have $ZF_1+ZF_2$ a constant. That means that
$$ AF_1+AF_2 = BF_1+BF_2 $$ $$ AF_1+AF_1+F_1F_2 = BF_2+F_2F_1+BF_2 $$ $$ 2AF_1 = 2BF_2 $$ $$ AF_1 = BF_2 $$