Points Defined by Adjacent Pairs of Digits of Reciprocal of 7 lie on Ellipse

Theorem
Consider the digits that form the recurring part of the reciprocal of $7$:
 * $\dfrac 1 7 = 0 \cdotp \dot 14285 \dot 7$

Take the digits in ordered pairs, and treat them as coordinates of a Cartesian plane.

It will be found that they all lie on an ellipse:


 * EllipseFromSeventh.png

Proof

 * EllipseFromSeventhSolution.png

Let us join the points:
 * $\left({1, 4}\right)$ to $\left({2, 8}\right)$
 * $\left({1, 4}\right)$ to $\left({5, 7}\right)$
 * $\left({1, 4}\right)$ to $\left({8, 5}\right)$


 * $\left({4, 2}\right)$ to $\left({2, 8}\right)$
 * $\left({4, 2}\right)$ to $\left({5, 7}\right)$
 * $\left({4, 2}\right)$ to $\left({8, 5}\right)$


 * $\left({7, 1}\right)$ to $\left({2, 8}\right)$
 * $\left({7, 1}\right)$ to $\left({5, 7}\right)$
 * $\left({7, 1}\right)$ to $\left({8, 5}\right)$