SOLUTION: I'm looking for an equation to produce coordinates for a square from two given diagonal vertices that works even with arbitrarily rotated squares. I haven't found a clear answer in

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Question 479447: I'm looking for an equation to produce coordinates for a square from two given diagonal vertices that works even with arbitrarily rotated squares. I haven't found a clear answer in any of my searches :(
Answer by solver91311(24713) (Show Source):
You can put this solution on YOUR website!

Place one endpoint of the diagonal at the origin and then the other end point is .

Convert to polar coordinates :

Since the diagonal of a square bisects a vertex angle of , each of the sides of the square which can be represented by a vector: must be offset in angle from the diagonal by a measure of .

Furthermore, if the diagonal of a square is of measure , then the side .

Hence, the polar coordinates of the side rotated counterclockwise from the diagonal are:

and

Which can then be converted back to rectangular coordinates by:

First look at

Then using the sum formula for sin:

Next, note that

So:

The sum formula for cos is:

And from that I'll let you derive that:

And then using the fact that the other side of the square is rotated clockwise from your given diagonal and that all sides of a square are identical in measure, you should be able to see that:

and

I'll let you look up the difference formulas and verify that:

John

My calculator said it, I believe it, that settles it