J. Leech's solution is the nine points whose coordinates are:-

(0,0),(182,0),(209,0),(391,0)

(120,0),(120,182),(120,209),(120,391),(60,195.5)

These points can obviously be rotated and rescaled without making any of
the distances irrational.

My contribution is to note that if you perform geometric inversion about any one of the points then the new 8 points plus the origin form a new nine point set. This follows from the standard cosine formula for a triangle

a^{2} = b^{2}+c^{2} - 2*b*c*cos(theta)

if this is applied to a triangle whose sides are rational then cos(theta)
must be rational.

If you divide both sides by b^{2}*c^{2}
it is evident that the triangle with sides 1/a and 1/b and the same angle theta
has a rational third side a/b/c.

Thus repeated applications of this process produce a family of related sets.

I wanted to find out whether repeated applications of this process produced
a finite or an infinite family (excluding rotations and rescalings).

I avoided rotations by defining the set of points as the 36 distances
between them and avoided rescalings by fixing one of the distances as 1.

I wrote Mathematica code, applying this process to Leech's set
and found that the family is finite with less than 3200 members. However
my method does not exclude reflections so that the real size is much less.
I found that one always returned to the original set after a chain of no more
than 4 inversions.

Investigations continue.

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