Equidistance of Ellipse equals Major Axis

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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.

$\blacksquare$


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