3. Postdoctoral Program - MSRI

Proof. For each 1 ≤ i < j ≤ 3 where xi + xj = g(mij), we have three equations:
x1 + x2 = g(m12)
x1 + x3 = g(m13)
x2 + x3 = g(m23)
This is equivalent to the following row reduced augmented matrix:
⎛
⎞
⎜ 1 0 0
⎜ 0 1 0
⎜
⎝
which gives us our desired values.
g(m12)+g(m13)−g(m23)
2
g(m12)−g(m13)+g(m23)
2
0 0 1 −g(m12)+g(m13)+g(m23)
2
We can follow a similar argument to find a rational distance set of four rational
points on a parabola.
Theorem 4. Given a parabola y = ax 2 + bx + c, let T = {P1, P2, P3, P4}. T is a
rational distance set of rational points if and only if there are rational values mij,
1 ≤ i < j ≤ 4 such that
x1 = 1
2 (g(m12) + g(m13) − g(m23))
x2 = 1
2 (g(m12) − g(m13) + g(m23))
x3 = 1
2 (−g(m12) + g(m13) + g(m23))
⎟
⎠
x4 = 1
2 (−g(m12) − g(m13) + g(m23) + 2g(m14))
and g(m13) + g(m24) = g(m23) + g(m14) = g(m12) + g(m34)
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