Construction of major and minor axes of an ellipse given only 2 focus points $(F_1,F_2)$ and a point $P$ that is on the ellipse.
Suppose we define $|F_1P|+|PF_2|:=l$
First I constructed the perpendicular bisector of $F_1F_2$ this is essentially the minor axis given the symmetry of the foci.
To obtain the boundaries, I was thinking of taking the segment $F_1P$ and extend it (Euclid's 2nd Axiom). Using a compass, with radius set to $|PF_2|$ and centred at $P$, draw a circle and mark the intersection with the extended line $F_1P$ as $F_1'$ where this point is on the opposite side of $F_1$ with respect to $P$. Now I construct the perpendicular bisection of $F_1F_1'$ to determine its midpoint (let's call it $M$).
Am I wrong to say that $M$ lies on both the minor axis as well as the ellipse?