Suppose that we have points $A$ and $B$, and the distance from $A$ to $B$ is $1$. Construct points $C$ and $D$ such that the distance from $C$ to $D$ is $\sqrt{2}$
So far I have found how to solve this problem when the line segment has length $2$. But in this case the length is $1$. So this is what I have done
Construct the line through $A$ and $B$. Next, construct a semicircle centred at $B$ with radius $AB$. Sat it meets he line at $A$ and $C$. Now consider the bisector of $A$ and $C$. This line passes through $B$ and intersects the semicircle at a point, $D$. Clearly, the length of the line segment $BD$ is $1$. Forming a right triangle of legs $BC$ and $BD$ and hypotenuse $CD$, the Pythagorean theorem would complete the construction. Am I right?
I need to know if there are any steps that are redundant or if any of them can be replaced by something more elemental. I would also like to know if there is another (elementary too) way to do this exercise.