If
Points A, B, C, D lie on a hyperbola, with AB parallel to CD, then
the line joining the mid-points of AB and CD passes through the
centre of the hyperbola.*

Choose any two points A and B on the curve, and find M, the
mid-point of AB. Construct a line parallel to AB that cuts the curve
at C and D, and find N, the midpoint of CD. Draw MN.

Now repeat the whole process starting with new points A’, B’, and
finishing with a line M’N’.

The centre, O, of the hyperbola is the intersection of MN and M’N’.

Draw a circle with centre at O which intersects the hyperbola at
P, Q, R, S. The two lines through O parallel to the sides of the
rectangle PQRS will be the two axes of the hyperbola.

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* Outline Proof

The line y = mx + c crosses the hyperbola x^{2}/a^{2} – y^{2}/b^{2}
= 1 at