Mohr-Mascheroni with collapsing compass

Question

By famous Mohr-Mascheroni theorem

Every geometric construction that can be carried out by compass and straightedge can be done with the compass only (without a straightedge).

To say in short, to prove the theorem, we have to prove that the three following constructions can be done with only compass:

1. Points of intersection of two circles given by center and one of the points for each circle
2. Points of intersection of a circle (given by center and one of its points) and a straight line (given by two points).
3. Point of intersection of two straight lines each of them given by two points.

I was reading "A short elementery proof of the Mohr-Mascheroni Theorem" by Norbert Hungerbuhler.

But it seems to me that the autor uses transport of the measure by the compass.

I suspect that we can avoid the usage of transport of measure by compass in the proof of Mohr-Mascheroni theorem. That is I do believe that every point constructible by collapsing compass and a straightedge can be constructed by means of collapsing compass only. But unfortunately I still find myself unable to do that.

P.S. It seems to me that despite the comments below, the construction in the Problem 4 of the book of Kostovskiy mentioned in the answer by @saltandpepper uses the measure tramsport as well [constructing the circles $(O,a)$, $(C, OE)$, $(D, OE)$].

Answer 1

The obstacle that bound me can be elininated by the elegant equivalence between two kinds of compasses a "rigid" one and a collapsing "divider".

Suppose we want to build a circle with center $A$ and the radius $BC$. Then we do the following: first draw blue circles, then red ones, then resulting green one; as it is shown below: [source: Wikipedia] (https://i.stack.imgur.com/aA1CA.png) Source of the answer is the underrated comment on my other old quaestion: https://math.stackexchange.com/a/2933016/239005

Answer 2

Why do you say that they require measurements by compass?

Is it because of the reflections? Observe that you can construct the reflection of $M$ with respect to the line passing through $P_1$ and $P_2$, by drawing the circle with center $P_1$ and radius $P_1M$ and drawing the circle with center $P_2$ and radius $P_2M$ you get two intersection points $M$ and $M'$.

Besides constructing reflections (which they claim but I didn't see proven) I didn't see any other step that can be seen as using measurements.

The midpoint construction is explicitly there on page $2$. Did I miss some other instance?

Doubling is also OK. Take into account that necessarily one should be allowed to, given two points, place the needles of the compass on each of them, otherwise we wouldn't be able to draw the circle with center on one of them and passing through the other.

This Mir book also has it: Geometrical Constructions Using Compasses Only by A. N. Kostovskii. The link has the translation in Spanish.

The first and second constructions in Kostovskii's book are precisely the reflection of a point, and multiplying a segment by an integer number.