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Constructing the Radius of a Ball

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 .

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Reference
[1] B. A. Kordemsky, The Moscow Puzzles, London: Penguin, 1990 pp. 251–252.
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