Given AB
Construct perpendicular to AB at B Prop 11
Construct circle with center B and radius AB Post 3
Extend AB Post 2
Let Z be the 2nd intersection of ray AB and the circle Def 17
Construct circles with centers A and Z and radii AZPost 3
Let X and Y be their intersections Assumption that 2 circles in a plane intersect in 2 pts
Construct XYPost 1
Notice that AX=XZ.They are radii of circles with radius AZ, Def 15, and CN 1.
Notice that XB=XB. CN4
Notice that AB=BZ. Def 15 as they are radii of the circle with center B and radius AB.
Since triangle ABX is congruent to triangle ZBX, then angle XBA= angle XBZ. Prop 8 (SSS)
Hence angles XBA and XBZ are right.Def 10
Construct C as the intersection of XB and the circle at B of radius AB.Def 15
Notice that BC=AB. Def 15
In addition, angle ABC is right Angle ABA is right (above), C is on XB, Def 8, CN4.
We have constructed a figure with 4 equal sides and 4 right angles. Thus we have constructed a square. Def 22

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