Feynman gave a geometric proof that planet's orbit around the sun is an ellipse, there's an youtube episode for that, and for the tricky part there's some discussion on stackexchange physics forum, a more formal explanation can be found from the paper Paths of the planets by R.H. Hall and N. Higson.
However, Hall etc's paper used a "Tangent Principle", which says :
If two curves (in polar coordinates) $r_1(θ)$ and $r_2(θ)$ have the same tangent at every $θ$, then they are the same, up to scaling.
I believe this is also implicitly used by Feynman. It sounds correct, however, how could one prove it?
Effect of scaling on tangent.
I think one would have to dilate both $x$ and $y$ axes the same amount for your argument. Consider the function $g(x,y) = (ax,by)$ , let us consider the position vector to the curve $\gamma(x,y)$ then in the scaled system it will be $g \circ \gamma(t)$. The tangent vector is given as:
$$ \frac{d}{dt} (g \circ \gamma(t) ) = \frac{d}{dt} ( a x(t), a y(t) ) = a \frac{d \gamma}{dt}$$
Hence, the tangent vector is parallel to the original under stretch/ squish. To recover the original statement that the curves will be the same upto scale, simply integrate the above equation.