### Duplicate the cube?

An idea for duplicating a cube. The problem is this. Given a cube of volume *a*^{2},
construct a cube of volume 2*a*^{2}. More precisely, construct a side *b* of such
a cube. In more modern terminology, give a length *a*, constuct a length *b* so that
*b/a* is the cube root of 2. If you set the length of *a* to be one, then the problem is
just to construct a line whose length is the cube root of 2. The limitation is that this must be done
with Euclid's tools of plane geometry, the straightedge and compass.
Here's one proposed solution. Construct an isosceles triangle with base *BC* equal to 1 and
height *AD* equal to 2.

Trisect the base *BC* at points *E* and *F*, that is, make
*BE* = *EF* = *FC*, each of length 1/3. Then connect *E* and
*F* to the vertex *A*.

Let *G* be the midpoint of the side *AB*. Construct an isosceles triangle *BGH*
with *H* a point on the other side *AC*, and *BG* = *BH*.

Let *GH* intersect *AE* at *L*, and *GH* intersect *AF* at *J*.
Also, let *BH* intersect *AE* at *O*, and *BH* intersect *AF* at *N*.

Then draw lines parallel to *GH* through *N* and *O*, and let them intersect
*AB* at *K* and *I*, respectively. Also draw lines *KL* and *IJ* parallel to
*BH*.

The three lines *IJ*, *AE*, and *KN* meet at a point. (They look like they
do,anyway.)

It was proved in the 19th century that a cube can not be duplicated with the Euclidean tools of
straightedge and compass.

This page:
June, 2001

David E. Joyce

Department of Mathematics and Computer Science

Clark University

Worcester, MA 01610

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