In Book II of Euclid's Elements, Proposition 11 reads:
To cut a given straigt line so that the rectangle contained by the whole and one of the segments is equal to the sqare on the remaining segment.
By this Euclid is proposing a method to construct a point H such that AB : AF = AF : FB. He then constructs figure A (see inset) as follows:
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From the given line AB, he creates the square ABCD. E is defined as the midpoint of the line AC. By use of a compass, the distance of EB is translated to the line EF passing through the point A. He then makes a square on AF, and calls the unlabeled point G. Finally, he draws a line from G through H and stops at its intersection with the line CD. This point of intersection is callek K. The rectangle AHKC is a golden rectangle. This discovery was not the goal of Euclid's proposition, but because the golden ratio satisfies his proposition, we now have a formal method for construction the golden ratio.
To prove it algebraically,
