Unit II Summary: Book I Deductive Arc (I.1 → I.47)

From Equilateral Construction to Pythagorean Area Decomposition

Book I ascends in three architectural stages, each illuminated by Byrne’s four-color register design:

1. Pure Incidence & Triangle Congruence (``I.1``–``I.26``): Without parallel lines, Euclid constructs the equilateral triangle (I.1), establishes SAS congruence (I.4), proves isosceles base-angle equality in the Pons Asinorum (I.5), bisects angles and segments (I.9–I.10), erects perpendiculars (I.11–I.12), and proves vertical-angle equality (I.15).

2. Parallel Transport & Angle Sum (``I.27``–``I.34``): Crossing the threshold of the Parallel Postulate (Simson Axiom XII / [I.post.5]) in I.29, Euclid constructs parallels (I.31) and proves in I.32 that the exterior angle of any triangle equals the sum of its two opposite interior angles—displaying the conservation of visually as adjacent colored wedges.

3. Parallelogram Shear & The Pythagorean Climax (``I.35``–``I.48``): Area equivalence under shear between parallel lines (I.35–I.41) and square construction (I.46) culminate in I.47, where thealtitude from the right angle splits the hypotenuse square into two colored rectangles individually equal to the squares on the adjacent legs.