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How did Euclid solve quadratic problems by applying areas?

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Three theorems of congruent triangles — Euclidean geometry This diagram depicts three basic theorems that triangles are congruent—that is, they are of equal shape and size—if: (1) two sides and the included angle are equal, designated SAS; (2) two angles and the included side are equal, designated ASA; or (3) all three sides are equal, designated SSS.

In the Elements, Euclid utilized a procedure known as the "application of areas" to solve quadratic problems through geometric means, a technique primarily detailed in Books I, II, and VI. This method, which can be associated with 5th-century BCE Pythagoreans, treats algebraic identities as theorems concerning equivalent geometric figures, often referred to as geometric algebra.12

The procedure involves finding a line segment $x$ that satisfies specific area relationships relative to given lines $a$ and $b$ :

Parabola (parabolē): Seeking $x$ such that $a x = b^{2}$ .
Hyperbola (hyperbolē): Seeking $x$ such that $x (a + x) = b^{2}$ , where the application is in "excess" by a square figure ( $x^{2}$ ).
Ellipse (elleipsis): Seeking $x$ such that $x (a - x) = b^{2}$ , where the application is in "defect" by a square figure ( $x^{2}$ ).

These constructions provided a rigorous geometric foundation for what would later be understood as the solution to general quadratic equations. While Euclid presented these as purely geometric propositions, later mathematicians, such as al-Khwārizmī, utilized these Euclidean proofs to provide geometric justifications for algebraic procedures.13