How did Euclid construct the division of a line later called the golden section?

In his seminal work Elements, Euclid described the division of a line segment into what he termed an "extreme and mean ratio," a concept now widely known as the golden section.

Euclid defined this division as a proportion where the ratio of the whole line segment to its larger part is equal to the ratio of the larger part to the smaller part. While the term "golden section" was not coined until the 19th century, Euclid provided the foundational geometric construction for this ratio in Book II of the Elements using classical methods, such as the straightedge and compass. This construction is a cornerstone of his work in plane geometry and remains a significant example of how ancient mathematicians approached proportions through geometric figures rather than modern algebraic notation.123