Using a ball, a sheet of paper, a compass, a straightedge without markings, and a pencil, can you draw a line segment on the paper equal to the ball's radius?

Step 1. Choose a point

on the ball and draw a circle with center

on the ball. Choose points

,

, and

on that circle.

Step 2. Draw a triangle congruent to

on the paper and construct the center

of its circumscribed circle.

Step 3. Draw two mutually perpendicular lines (horizontal and vertical) through

. Let

be the intersection of the horizontal line with the circle. Find the point

on the vertical line such that the segments

and

are equal. (

is also equal to the segments

and

.)

Step 4. Find the intersection

of the vertical line and the line perpendicular to

through

.

If the point

is opposite

on the ball, then triangle

is congruent to

. So the segment

equals the diameter of the ball.

Step 5. To get a segment equal to the ball's radius, construct point

, the midpoint of segment

.