Get ready Part 18

Suppose that four points in $\mathbb{R}^2$ are such that any two of them are an odd integer distance apart. If we make one of these points the origin, we have three vectors $\mathbf{a}$ , $\mathbf{b}$ , $\mathbf{c}$ such that $\Vert\mathbf{a}\Vert$ , $\Vert\mathbf{b}\Vert$ , $\Vert\mathbf{c}\Vert$ , $\Vert\mathbf{a-b}\Vert$ , $\Vert\mathbf{a-c}\Vert$ , $\Vert\mathbf{b-c}\Vert$ are all integers. But then $\Vert\mathbf{a}\Vert^2$ , $\Vert\mathbf{b}\Vert^2$ , $\Vert\mathbf{c}\Vert^2$ , $\Vert\mathbf{a-b}\Vert^2$ , $\Vert\mathbf{a-c}\Vert^2$ , $\Vert\mathbf{b-c}\Vert^2$ are all integers congruent to $1$ modulo $8$ . But this implies that $2\mathbf{a}\cdot\mathbf{b} \; = \; \Vert\mathbf{a}\Vert^2 + \Vert\mathbf{b}\Vert^2 - \Vert\mathbf{a-b}\Vert^2$ and similarly $2\mathbf{a}\cdot\mathbf{c}$ and $2\mathbf{b}\cdot\mathbf{c}$ are integers congruent to $1$ modulo $8$ .

Consider the $2\times 3$ matrix $X$ whose columns are $\mathbf{a},\mathbf{b}, \mathbf{c}$ . Then $X$ has rank at most $2$ , and hence the $3\times3$ matrix $Y= 2X^TX$ has rank at most $2$ , and hence is singular, and so $\mathrm{det}\,Y = 0 \equiv 0 \pmod{8}$ . But $Y \; = \; 2\left(\begin{array}{ccc} \Vert\mathbf{a}\Vert^2 & \mathbf{a}\cdot\mathbf{b} & \mathbf{a}\cdot\mathbf{c} \\ \mathbf{a}\cdot\mathbf{b} & \Vert\mathbf{b}\Vert^2 & \mathbf{b}\cdot\mathbf{c} \\ \mathbf{a}\cdot\mathbf{c} & \mathbf{b}\cdot\mathbf{c} & \Vert\mathbf{c}\Vert^2 \end{array}\right) \equiv \left(\begin{array}{ccc} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{array}\right) \hspace{1cm} \pmod{8}$ which implies that $\mathrm{det}\,Y \; \equiv \; 4 \pmod{8}$ which is a contradiction. Thus no such four points can exist.

We call 4 points are Ax1y1, Bx2y2, Cx3y3, Dx4y4.

Distance from points odd when only when xi-xj and yi-yj have different parity Consider x1-x2, x1-x3, x1-x4; there were 1 pair have the same parity. Same as y1-y2, y1-y3 and y1-y4 but having the reverse.

So in general we can have:

X1-x2 and x1-x3 are even, y1-y2 and y1-y3 are odd. x1-x4 is odd, y1-y4 is even.

Thus x3-x4 is odd and y3-y4 is odd. So distance from DC is even number.