Euclid Challenge - Squaring A Circle By Straightedge And Compass

Successful Response by Milton Mintz

May 10, 2002

Page 11: Squaring A Circle By Straightedge And Compass

Converting the area of the Circle -- π(1)² -- to a Rectangle

Draw line XY.
Draw line GH perpendicular to line XY, intersecting line XY at point B.
Center, point B, radius 1, draw circle, intersecting line XY at points K and L, and line GH at point D.
Bisect arc KD, mark point O. Bisect arc KO, mark point P.
Right of point L, draw line G’H’ perpendicular to line XY, intersecting line XY at point B’.
Grasp with the left hand a compass leg above the legpoint. (Identify as leg R).
Place legpoint of leg R on point K.
Grasp with the right hand the other compass leg above the legpoint. (Identify as leg S).
Place legpoint of leg S on point B’.
Move both legpoints at a Uniform Rate; legpoint of leg R along arc KD towards point D; and legpoint of leg S along line B’H’ towards point H’.
When legpoint of leg R reaches point P, stop both legpoints. At the location of the legpoint of leg S, mark point P’.
Line B’P’= 1/8 of the ACTUAL length of π.
Draw line X’Y’.
On line X’Y’, from point X’, length line B’P’, mark point P’.
On line X’Y’, from point P’, at end of 7 additional lengths of line B’P’, mark point W.
X’W = ACTUAL length of π , approximate numerical value = 3.141592654.
Draw line (a)(b).
On line (a)(b), mark line X’W.
From point X’,draw a perpendicular line, length of radius (1), to point M.
Draw line MN parallel to line (a)(b).
From point W, draw a perpendicular line WW’, intersecting line MN at point C.

The area of the Circle -- π(1)² -- has been converted to the Rectangle (X’W)(X’M).