Unit VI: Book VI — Similar Figures & Extreme and Mean Ratio¶
Proportional Geometry & Application of Areas (Pickering pp. 213–268, Leaves 247–302)
Book VI Key Definitions (p. 213)¶
Definition I. Similar rectilineal figures are those which have their several angles equal, each to each, and the sides about the equal angles proportionals (∷).
Definition III. A straight line is said to be cut in extreme and mean ratio (Golden Section) when the whole line is to the greater segment as the greater segment is to the less.
Key Theorems of Similarity (VI.1–VI.13)¶
Prop. I–II (``G.286``–``G.287``): Triangles of the same altitude are to one another as their bases; a line drawn parallel (∥) to one side of a triangle cuts the other two sides proportionally, and conversely.
Prop. III (Angle Bisector Theorem, p. 216 · ``G.288``): Bisecting an angle of a triangle divides the opposite base into segments proportional to the adjacent sides.
Prop. XIII (Mean Proportional · ``G.298``): Constructing the geometric mean $\sqrt{ab}$ between two segments via Thales’s semicircle (
III.31).
Proposition XXX. Problem (Extreme & Mean Ratio, p. 258 · G.328)¶
To cut a given finite straight line ( + ) in extreme and mean ratio.
Construction (:just:`I.46`, :just:`VI.29`). On the given straight line describe the unit square ( + ), and to the vertical side apply a rectangle equal to the whole unit square and exceeding by a figure () similar to the square (VI.29).
Demonstration (:just:`C.N.3`, :just:`VI.14`).
Since the applied rectangle ( + ) equals the unit square ( + ), subtracting the common yields:
= .
Because equal equiangular parallelograms have reciprocally proportional sides ([VI.14]),
∴ whole line : ∷ : . Q.E.F.
Proposition XXXI (Generalized Pythagorean Theorem, p. 259 · G.329)¶
Immediately following VI.30, Euclid generalizes I.47 from squares to any three similar and similarly described rectilineal figures on the sides of a right-angled triangle—completing the grand arc of Books I–VI.