If a straight line be cut into two equal parts and also into two unequal parts, the rectangle contained by the unequal parts together with the square on the line between the points of section is equal to the square on the half.
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"Let it be granted that all right angles are equal to one another."
"A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line."
"To construct an equilateral triangle on a given finite straight line."
"And when the lines containing the angle are straight, the angle is called rectilineal."
"The square on the side subtending the right angle in right-angled triangles is equal to the squares on the sides containing the right angle."
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This describes what modern algebra writes as (a+b)(a-b) + b² = a². A line divided at its midpoint and also at a second point creates a rectangle from the two unequal segments and a square from the gap between the two cut-points. Their combined areas always equal the square built on the half-line. Euclid proves an algebraic identity through pure geometry, using only areas and lengths, with no symbols or variables.
Euclid (~300 BCE) systematized Greek mathematics in the Elements into rigorous logical proofs. This proposition reflects his core method: expressing what we now call algebraic identities purely as geometric area relationships, because symbolic algebra did not yet exist. His teaching in Alexandria and the famous remark that there is no royal road to geometry confirm his conviction that mathematics demands careful, stepwise reasoning — exactly what this meticulous proposition embodies.
Around 300 BCE, Alexandria under Ptolemy I was the Mediterranean world's intellectual hub, home to its great Library. Mathematics was entirely geometric — Babylonians had solved quadratic problems using area models, and Greeks formalized this into proof. No symbolic algebra existed, so relationships between quantities had to be demonstrated through shapes. This proposition gave architects, astronomers, and engineers a rigorous tool for solving proportion and measurement problems central to construction and commerce.
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