Constructing the Cube Root of Two

This Demonstration shows a construction of using a ruler with two marks (red and black ) at a distance 1 unit apart, constrained to lie on the axis and the blue line. Drag point until the straight line touches the hexagon at . Then .
Explanation: The slope of is and the slope of the blue line is . Since their product is , the blue line is perpendicular to the line . Therefore is a right-angled triangle, so . The angle is , so . Since triangles and are similar, . Therefore , substitute for to get , and clear the denominator: . Expand to get , rearrange terms to get , and factor , ; and take the positive solution .

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Reference
[1] J. H. Conway and R. K. Guy, The Book of Numbers, New York: Copernicus Books/Springer, 2006 pp. 194–195.
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