Points Defined by Adjacent Pairs of Digits of Reciprocal of 13 lie on Hyperbola

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Theorem

Consider the digits that form the recurring part of the reciprocal of $13$:

$\dfrac 1 {13} = 0 \cdotp \dot 07692 \dot 3$

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

It will be found that they all lie on a hyperbola:


HyperbolaFromThirteenth.png


Proof

HyperbolaFromThirteenthSolution.png


Let the points be labelled to simplify:

$A := \left({0, 7}\right)$
$B := \left({7, 6}\right)$
$C := \left({6, 9}\right)$
$D := \left({9, 2}\right)$
$E := \left({2, 3}\right)$
$F := \left({3, 0}\right)$



Sources