The basic constructions Compass-and-straightedge construction

the basic constructions

all compass , straightedge constructions consist of repeated application of 5 basic constructions using points, lines , circles have been constructed. these are:

creating line through 2 existing points
creating circle through 1 point centre point
creating point intersection of 2 existing, non-parallel lines
creating 1 or 2 points in intersection of line , circle (if intersect)
creating 1 or 2 points in intersection of 2 circles (if intersect).

for example, starting 2 distinct points, can create line or either of 2 circles (in turn, using each point centre , passing through other point). if draw both circles, 2 new points created @ intersections. drawing lines between 2 original points , 1 of these new points completes construction of equilateral triangle.

therefore, in geometric problem have initial set of symbols (points , lines), algorithm, , results. perspective, geometry equivalent axiomatic algebra, replacing elements symbols. gauss first realized this, , used prove impossibility of constructions; later did hilbert find complete set of axioms geometry.