Oops, Square roots!
October 11, 2011 2 Comments
It was pointed out yesterday that I forgot to prove, or even mention the fact that is closed under taking square roots. That is, if , then so is . So here goes.
Suppose we can construct . Build a line segment of length . Construct an additional segment of length 1 after it, so we have a segment of length . Now draw a circle with this as the diameter. You should have something like this:
Great! Now we can drop a perpendicular at , and see where it intersects the circle. Call this point . Remember from my first post on constructibility that we can construct perpendiculars through a point. We now have:
We can then call on some altitude on hypotenuse theorems. Namely, , so But in our diagram, , and , so , as desired.