suppose we have already drawed the perpendicular to the three parallel lines.

now point the drawing compass in Q with aperture QR and rotate until the compass cuts line q point, say, A.

the segment AS is a side of a square. report AS with the compass and rotate until cuts line r in point, say, B. AS in perpendicular to aB.

repeat the same procedure on the other end of the perpendicular, i.e. in S taking SR with the compass and

cutting line s in a point, and going on with the same operations as for the previus case-

Prob. 3 You must imgine the cube inscribed in a sphere. only the 8 vetices touch the sphere.

now think about the diagonal as an a rotation axis and a plne passing through it. let rotae the spere

together with the cube. you see the shape of a section of our cube rotaing, but the angle is the smallest when a vertex touches the sphere (90 degree). so only from 6 points see the diagonal under the smallest angle.

(r is between q, s)

Now

take on q a segment QA = RS and

take on S a segment SB = QR

on the same side of p.

angle ARB measures 90 by construction. the fourth vertex C is found in the same way.

the required square is the one inscribed in a (bigger) square with side QS ]]>

so that R = H_a/3 ]]>

Creating a problem that has a unique solution (like the one above) is less obvious, or isn’t it ? ]]>