Equivalence of focus-focus and focus-directix definitions of ellipse without leaving the plane

Question

Take a look at the following two definitions of ellipse:

For some fixed points $F_1,F_2$ and real number $2a>|F_1F_2|$ an ellipse is the locus of points $P$ such that $|F_1P|+|F_2P|=2a$.

For some fixed point $F$, line $d$ and number $e<1$ an ellipse is the locus of points $P$ such that $|FP|$ is $e$ times the distance from $F$ to $d$.

These two definitions can be easily shown equivalent using Dandelin spheres (which, in fact, also estabilishes that ellipse can be defined as a kind of conic section). However, for some time, I have been wondering if there is a way to show these definitions equivalent while "staying on the plane", i.e. without Dandelin spheres, cones etc.

My question here is: Is there any direct proof of equivalence of the above two definitions of ellipse? Also with "direct" I mean one without using the equation of an ellipse.

Thanks in advance.

Answer 1

$\frac{r1}{d1} =\frac{r2}{d2} = \frac{r1+r2}{d1+d2} = e$; Obviously $d_1+d_2$ is constant, so r1+r2 is also constant