I am just wondering if this is an axiom:
For a square, in order to draw a straight line between opposite diagonals, the line needs to pass through the mid-point.
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We must define the center of a square.
And if we define it as the unique point that is equidistance from the four vetices then we must first have a theorem that such a point exist and is unique.
And after that we can prove the diagonals pass through that point.
In standard Euclidean geometry we can prove it more or less like the following:
The set of all points equidistance from two opposite vertices is, by definition, the perpendicular bisector of of the diagonal. So the any point that is equidistance from all four points would be the point of intersection of the two perpendicular bisectors of the the diagonals. As two lines intersect at only one point this point is unique.
We can prove that the point of intersection of the two diagonals is equidistance from the four vertices (next paragraph). So the point of intersection of the diagonals is this point.
A diagonal cuts a square into two triangles triangles. As its a square these triangles have equal corresponding sides and they are congruent. And thus the base angles are equal (and they bisect a right angle and so are 45 degrees). So the two diagonals cut the square into four triangles. By symmetry and angle and component chasing it is easy to prove the four triangles are congruent. So the point of intersection is equidistance from the four vertices.
ANd that's that. In defining the center of a square as the unique point equidistance from all vertices, and in proving such a point exists and is unique, we proved that the diagonals intersect at the center.
No, we needn't include that as an axiom because it's a theorem; I'll discuss just one way to prove it. If you give the vertices the usual Cartesian coordinates, one diagonal is $y=x$ while the other is $y=1-x$, so they both pass through the centre of the square, at $(1/2,\,1/2)$.