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 AZ Post 3 Let X and Y be their intersections Assumption that 2 circles in a plane intersect in 2 pts Construct XY Post 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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