friends! Hilbert proves in the Grundlagen der Geometrie (ed. By Paul Bernaus) that the following problems can be solved using straightedge and fixed compass, with constant radius:
- to join two points with a straight line and to find the intersection between two non-parallel straight lines;
- to lay off a given segment on a given straight line from a given point on a given side of the line;
- to lay off a given sangle on a given straight line from a given point on a given side of the plane;
- to draw from a given point $A\notin a$ the parallel to a given straight line $a$;
- to draw a perpendicular to a given line from a given point.
I read, then, that a geometrical problem [...] solvable by drawing straight lines and laying off segments, i.e by using a straight edge and a fixed compass as if the two things were equivalent... Is that so? Given a circle of radius $r$ and a straight line, or given two circles of equal radius $r$, is it possible to find the intersection points by using only the constructions 1-5? We obviously can draw the perpendicular from the center of a circle to a given straight line, and to draw the perpendicular to the line $\overline{C_1 C_2}$ joining the centers $C_1$ and $C_2$ of two circles of equal radius $r$ from the middle point of the segment $C_1 C_2$ (the midpoint can be found by transferring segments and angles, according to theorem 26 of the Grundlagen, 1968 ed. by Bernays), but then, I wouldn't be able to find the cathetus and hypotenuse whose common point is the intersection between the circle and line or, respectively, two circles.
I $\infty$-ly thank you in advance for any help!