About a given circle to circumscribe a square be the given
circle; thus it is required to circumscribe a square about the
circle ABCD Let two diameters AC of the circle ABCD be drawn at
right angles to one another, and through the points A be drawn
touching the circle ABCD . [III. 16, Por. Then, since FG touches
the circle ABCD , and EA has been joined from the centre E to
the point of contact at A , therefore the angles at A are right.
[III. 18] For the same reason the angles at the points B, C, D
are also right. Now, since the angle AEB is right, and the angle
EBG is also right, therefore GH is parailel to AC. [I. 28] For
the same reason AC is also parallel to FK, so that GH is also
parallel to FK. [I. 30] Similarly we can prove that each of the
straight lines GF, HK is parallel to BED. are parallelograms;
therefore GF is equal to HK , and GH . [I. 34 And, since AC is
equal to BD , and AC is also equal to each of the straight lines
GH , while BD is equal to each of the straight lines GF, HK, [I.
34] therefo