TITLE: Need help understanding the proof regarding the locus of the feet of the perpendiculars from the focus of an ellipse to all of it's tangents
QUESTION [0 upvotes]: From page 12-13 of Arnold's "Real Algebraic Geometry"

I understand the following perfectly well. I included here because it is referenced in the next proof, which I need help with.



Now lets move onto the problem/proof that I need help understanding:
Problem: Find the locus of the feet R of the perpendiculars drawn from the focus F2 of an ellipse to all of its tangents.
(Skipping the proof for degenerate cases...._



Here is the part I wish to understand better:
"Therefore the locus of R is obtained by the locus of $F_2'$ as the result of a double contraction to the point $F_2$"
The next paragraph expands on what happens under this double contraction. I get quite lost trying to understand what is happening here. If somebody could explain to me what's going on, I'd greatly appreciate it. Thank you.

REPLY [1 votes]: Consider the midpoint $A$ of $F_1F_2'$: its locus is a circle $c$ of centre $F_1$ and radius $a$. If $R$ is the midpoint of $F_2F_2'$, then $AR$ is parallel to $F_1F_2$ and half its length. Hence the locus of $R$ is the translated of circle $c$ by a vector $\overrightarrow{F_1O}$, i.e. a circle of centre $O$ and radius $a$.
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