TITLE: A construction using straightedge and compass
QUESTION [6 upvotes]: Given a circle, it's easy to contruct its center.
The question is: given an ellipse, draw the foci.
I don't know whether it's possible to do this using only straightedge and compass.

REPLY [13 votes]: Let $\lambda$ a given ellipse, let's draw its foci.
See the figure below:


Draw two parallel chords $AB$ and $DC$.
Find out $M$ and $N$ (midpoints of $AB$ and $DC$ respectively).
Draw the straightline $r$ through $M$ and $N$.
Let $\{E, E'\} = r \cap \lambda$, find out $O$ the midpont of $EE'$.
Draw an arc $\mu$ centered at $O$ and radius $AO$, such that $\{A, A'\}= \mu \cap \lambda$.
Draw the straightline $s$ through $A$ and $A'$.
Draw a straightline $t$ such that $O \in t$ and $t\parallel s$.
Draw $u$ (the perpendicular bisector of $AA'$).
Let $\{G, G'\} = u \cap \lambda$, and $\{J, J'\} = t \cap \lambda$, $GG'$ and $JJ'$ are the axes of symmetry of $\lambda$. Now draw an arc $\nu$ centered at $G$ and radius $JO$, such that $\{F1, F2\}= \nu \cap t$. $F1$ and $F2$ are the desired foci of $\lambda$.