copying a segment

You have a segment of some length sitting in one corner of the page, and you want an identical copy starting at a chosen point somewhere else, pointing in a chosen direction. With a ruler you would just read off the centimetres and remeasure — but the classical game forbids measuring. The question is how to reproduce a length using only an unmarked straightedge and a compass that may even be collapsing.

Here is the idea with a rigid compass, which is how it is normally drawn. Given segment AB and a target point C, draw the line through C in the direction you want. Open the compass to the gap |AB|, place the point at C, and swing an arc that crosses that line at a point D. Then |CD| = |AB|, so CD is your copy. With a collapsing compass it takes a few more arcs (Euclid I.1 builds an equilateral triangle first to anchor the transfer), but the result is identical.

This is the bedrock construction: almost every later figure relies on being able to lay down a known length wherever you need it. Building a triangle from three given side lengths, marking equal steps along a line, copying-an-angle — all of them quietly call on copying a segment. It is the constructive version of the everyday act of measuring, but done with geometry instead of a marked scale.