Algebra - Construct circle with radius AB around distinct point C

Question

I'm currently reading Antoine Chambert-Loirs "A field guide to algebra". The very start of the book is dedicated to the topic of Construction with ruler and compass.

The most import definition used:

Let $\Sigma$ be a set of points in the plane $R^2$. One says that a point $P$ is constructible with ruler and compass from $\Sigma$ if there is an integer $n$ and a sequence of points $(P_1,...,P_n)$ with $P_n = P$ and such that for any $i \in {1,...,n}$, denoting $\Sigma_i := \Sigma \cup \big\{ P_1,...,P_{i−1} \big\} $, one of the following holds:

• there are four points $A, B, C$ and $D \in \Sigma_i$ such that $P_i$ is the intersection point of the two nonparallel lines $(AB)$ and $(CD)$;

• there are four points $A, B, C$, and $D \in \Sigma_i$ such that $P_i$ is one of the (at most) two intersection points of the line $(AB)$ and the circle with center $C$ and radius $CD$;

• there are four points $O, M, O$ and $M \in \Sigma_i$ such that $P_i$ is one of the (at most) two intersection points of the distinct circles with, respectively, center $O$ and radius $OM$, and center $O$ radius $OM$ .

Note: The definition above does not include the creation of a new point via the intersection between the circle of radius $AB$ around a (from $A, B$ distinct) point $C$ and a line/different circle.

The author asks the reader to explain, how this type of creation of new points is possible, with just the three given moves.

At this point I'm stuck and have no clue how to construct these kind of points. Any ideas?

Answer 1

For short, the allowed moves are

1. Intersecting two lines through two pairs of given points;
2. Intersecting a line through two given points with the circle through a third point with a known center;
3. Intersecting two circles with known centers through two given points.

We may use two moves of type-3 to draw the midpoint and the perpendicular bisector of a segment, hence four moves of type-3 and one move of type-1 are enough to find the circumcenter and circumcircle of a triangle. Moreover, we may find the symmetric of $C$ with respect to the midpoint of $AB$ with two moves of type-3, one move of type-1 and one move of type-2:

• We consider the circles centered at $A$ and $B$ with radius $AB$: their radical axis meet $AB$ at its midpoint $M$;
• The $CM$ line and the circle with center $M$ and radius $CM$ meet at $C$ and at the symmetric of $C$ with respect to $M$, say $D$.

$ADBC$ is a parallelogram, hence the circle with center $A$ and radius $AD$ solves the problem.