# Intersection of a circle and a line with only compasses

Recently I asked a question about Mohr-Mascheroni theorem. I read the paper "A short elementary proof of the Mohr-Mascheroni Theorem" by Norbert Hungerbuhler but was unsatisfied with it.

In Construction 1 the author constructs intersection points of a circle with center $$M$$ and line non-passing a center of the circle given by two points.

The Construction 1 is as follows:

If the straight line $$g$$ is given by the points $$P_1$$ and $$P_2$$ we reflect the center $$M$$ of given circle $$K$$ with respect to $$g$$. (It is done by means of two circles one with center in $$P_1$$ and second with center in $$P_2$$ through $$M$$) Then we find the two points of intersection $$\{X,Y\}=K\cap g$$ as the point of intersection of $$K$$ and the reflected circle $$K'$$.

As the user @Aretino indicated in comment on the recent question Euclid's compass could only draw a circle given its center AND a point of the circle, which was my point when I asked the recent question about an reference of alternative proof.

So my question is: How to construct $$\{X,Y\}$$ as above with "Euclid's compass"?

I can't understand if construction of reflected circle is carried out in a correct way in the construction mentioned above, we have a center $$M'$$ of reflected circle, but haven't a point constructed to draw it through.

May be I misunderstood a Mohr-Mascheroni theorem, may be it is not about an "Euclide's compoass"?

I agree that the construction is unclear, though it is easily fixed. We just need to reflect any point on circle $$K$$ through $$g$$. For example:
1. The point $$Q$$ is already constructed as the intersection of $$K$$ with the circle with centre $$P_1$$ through $$M$$.
2. Reflect $$Q$$ through $$g$$ to $$Q'$$ using the already given point reflection construction. (Circles with centres $$P_1$$ and $$P_2$$ through $$Q$$, not shown in diagram.)
3. Construct $$K'$$ with centre $$M'$$ through $$Q'$$.