text: how do you inscribe a reular square in a circle? how would you inscribe an octagon in a circle? to incribe a regular quadrilateral (ie sqare) into a circle with only a compass and straight edge do the following:  1. draw a circle. label the center a.  2. draw a diameter of the circle by placing your straight edge on the center of the circle. label the points where the diameter intersects the circle as b and c.  3. set your compass to a wider amount than you used to draw your circle.  4. without adjusting your compass, draw a circle with b as the center.    5. without adjusting your compass, draw another circle with c as the center.   6. your two new circles will intersect at two points. use your straight edge to draw a line through these two points.  7. this new line will intersect the original circle at two points. label those points as d and e.   8. b, c, d, and e are the corners of your inscribed square. you can use your straight edge to draw the edges of the square.  reason this works:  in steps 4 through 6, you contructed the perpendicular bisector to the diameter of the original circle.   this means that segment de is a diameter of the orignal circle (since it passes through the center of the circle). this also means segment bc and de form a right angle.   if you look at triangles bad and dac; ba, ad, ac are all radii of the original circle (and thus the same length). therefore the triangles are exact copies of each other and bd and dc are the same legth. this logic will show you that all the sides are the same length.  angle dbe inscribes one half of the original circle since de is the diameter of the circle. this makes dbe a right angle. similar logic will show you all angles of your square are right angles.   since your sqare has all equal sides and all equal angles, it is a square.