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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3$\begingroup$ No.$\mathstrut$ $\endgroup$ – Ivan Neretin Jul 20 '20 at 16:52
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$\begingroup$ That fact follows from other things $\endgroup$ – Milan Jul 20 '20 at 16:54
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$\begingroup$ Why is it not though? $\endgroup$ – Prince Deepthinker Jul 20 '20 at 16:54
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$\begingroup$ How do you prove it $\endgroup$ – Prince Deepthinker Jul 20 '20 at 16:54
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$\begingroup$ What the heck is a midpoint of a square? That is .... what is a reasonable definition? $\endgroup$ – fleablood Jul 20 '20 at 17:03
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It could be an axiom if you really wanted it to. You can make your axioms be whatever you want. But I have never seen any conventional treatment of geometry with such an axiom.
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$\begingroup$ I agree. It COULD be an axiom of some form of geometry but it is not an axiom of Euclidean Geometry or any of the well known non-Euclidean geometries. $\endgroup$ – Simon Terrington Jul 20 '20 at 17:01
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$\begingroup$ I was having Euclidean geometry in mind $\endgroup$ – Prince Deepthinker Jul 20 '20 at 17:13
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1$\begingroup$ @Deepthinker101 Again, you could probably include this as an axiom in a geometry of your own making, and then prove that your geometry ultimately becomes in Euclidean geometry as we all know it. There are, after all, many ways to axiomatize Euclidean geometry, some more popular than others. But as I said, I have never seen your statement as an axiom. I don't think there is any good reason to have it as an axiom. It's better to just use a conventional axiom set and leave your statement as a theorem. $\endgroup$ – Arthur Jul 20 '20 at 17:18
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2$\begingroup$ "Wow I didn't know I could do such things in maths" Well, someone has to. Math didn't just fall from the sky. Someone had to invent it. And if s/he could we can too. $\endgroup$ – fleablood Jul 20 '20 at 17:23
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1$\begingroup$ Yes, but we have to interpret them in a language of our devising. If two "atoms" of meanings are equivalent (we can prove one from the other) which one we choose to be the axiom and which one we choose to be the result is entirely up to us. $\endgroup$ – fleablood Jul 20 '20 at 17:58
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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.
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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.
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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 $(1/2,\,1/2)$.
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$\begingroup$ What's the definition of a center of a square? And do centers exist for squares not placed and the origin of the coordinate system. And is accepting that Cartesian coordinates are valid an axiom? Which are the axioms and which are the conclusions? $\endgroup$ – fleablood Jul 20 '20 at 17:11
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$\begingroup$ @fleablood The path from axioms to where we want to go passes through many theorems and definitions, we can do it in various equivalent ways, and frankly there may be a more direct solution than proving we can add coordinates like this. But that's kind of the point: we're definitely not ending up at a standard axiom. $\endgroup$ – J.G. Jul 20 '20 at 18:06
