Equidistance of Ellipse equals Major Axis

Theorem
Let $K$ be an ellipse whose foci are $F_1$ and $F_2$.

Let $P$ be an arbitrary point on $K$.

Let $d$ be the constant distance such that:
 * $d_1 + d_2 = d$

where:
 * $d_1 = P F_1$
 * $d_2 = P F_2$

Then $d$ is equal to the major axis of $K$.

Proof

 * EllipseEquidistanceMajorAxis.png

By the equidistance property of $K$:
 * $d_1 + d_2 = d$

applies to all points $P$ on $K$.

Thus it also applies to the two vertices $V_1$ and $V_2$:


 * $V_1 F_1 + V_1 F_2 = d$
 * $V_2 F_1 + V_2 F_2 = d$

Adding:


 * $V_1 F_1 + V_2 F_1 + V_1 F_2 + V_2 F_2 = 2 d$

But:
 * $V_1 F_1 + V_2 F_1 = V_1 V_2$
 * $V_1 F_2 + V_2 F_2 = V_1 V_2$

and so:
 * $2 V_1 V_2 = 2 d$

By definition, the major axis is $V_1 V_2$.

Hence the result.