I am facing a simple problem in straightedge and compass construction, even for simple kind of real numbers.
Rules: A point $P$ in $2$-D plane or $\mathbb{C}$ is constructible by straightedge and compass from points $P_0,P_1,\ldots, P_n$ if
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$P$ is different from them;
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$P$ lies on intersection of two non-parallel distinct lines, $l_1,l_2$ which are obtained by joining some points in $P_0,\ldots, P_n$.
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$P$ lies on intersection of a line $l$ (obtained as in 2) and circle $C$ with center some $P_i$ and passing through some $P_j$.
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$P$ lies on two different intersecting circles $C,C'$ which are obtained in 3.
Starting with given points $0$ and $1$, I was able to show the construction of all rationals along with $\sqrt{2}$.
Question: I am facing problem of showing construction of $1+\sqrt{2}$.
Where is problem: One may say that,
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locate the point $\sqrt{2}$ on x-axis (this is okay!), denote it by $P$;
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Take unit distance to the right from $P$. [HOW?]
I am not able to convince myself:
if $A,B$ are constructed points, we can keep one end of compass at $A$, stretch other end up to $B$ and use this distance $d$ to draw circles of radius $d$ from other points!
Can one clarify this simple point? What is lack in my understanding?
Best Answer
If you place one end of the compass at A and the other at B, you are allowed to draw either a circle centered at A or a circle centered at B - nothing else. You may be thinking that you can lift the compass off the plane and transfer it to another point while retaining the separation of the two points of the compass by the distance d. In classic straightedge-and-compass constructions, this is explicitly not allowed - the compass is considered to be 'collapsible', i.e the joint where the two arms of the compass meet is 'loose', so the moment you lift the compass, the plane is no longer able to hold the two arms at distance d.
In the case at hand: you can draw a line bisecting the angle between the X and Y axes, and mark the point P' where it intersects the circle about the origin of radius $\sqrt2$. It is not hard to see that this point is at a distance 1 from the X and Y axes, so you can find a point of distance 1 on (say) the Y axis by drawing a line parallel to the X axis through P'. You can then draw a circle of radius 1 centered at P', intersecting the first line at P''. A circle centered at the origin through P'' will then intersect the X axis at a distance $1 + \sqrt2$ from the origin.