Book:Euclid/The Elements/Book VI

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Euclid: The Elements: Book VI

Published $\text {c. 300 B.C.E}$


Contents

Book $\text{VI}$: Theory of Proportions as applied to Plane Geometry

Definitions
Proposition $1$: Areas of Triangles and Parallelograms Proportional to Base
Proposition $2$: Parallel Transversal Theorem
Proposition $3$: Angle Bisector Theorem
Proposition $4$: Equiangular Triangles are Similar
Proposition $5$: Triangles with Proportional Sides are Similar
Proposition $6$: Triangles with One Equal Angle and Two Sides Proportional are Similar
Proposition $7$: Triangles with One Equal Angle and Two Other Sides Proportional are Similar
Proposition $8$: Perpendicular in Right-Angled Triangle makes two Similar Triangles
Porism to Proposition $8$: Perpendicular in Right-Angled Triangle makes two Similar Triangles
Proposition $9$: Construction of Part of Line
Proposition $10$: Construction of Similarly Cut Straight Line
Proposition $11$: Construction of Third Proportional Straight Line‎
Proposition $12$: Construction of Fourth Proportional Straight Line
Proposition $13$: Construction of Mean Proportional‎
Proposition $14$: Sides of Equal and Equiangular Parallelograms are Reciprocally Proportional‎
Proposition $15$: Sides of Equiangular Triangles are Reciprocally Proportional
Proposition $16$: Rectangles Contained by Proportional Straight Lines
Proposition $17$: Rectangles Contained by Three Proportional Straight Lines
Proposition $18$: Construction of Similar Polygon
Proposition $19$: Ratio of Areas of Similar Triangles
Porism to Proposition $19$: Ratio of Areas of Similar Triangles
Proposition $20$: Similar Polygons are composed of Similar Triangles
Porism to Proposition $20$: Similar Polygons are composed of Similar Triangles
Proposition $21$: Similarity of Polygons is Equivalence‎ Relation
Proposition $22$: Similar Figures on Proportional Straight Lines
Proposition $23$: Ratio of Areas of Equiangular Parallelograms
Proposition $24$: Parallelograms About Diameter are Similar
Proposition $25$: Construction of Figure Similar to One and Equal to Another
Proposition $26$: Parallelogram Similar and in Same Angle has Same Diameter
Proposition $27$: Similar Parallelogram on Half a Straight Line
Proposition $28$: Construction of Parallelogram Equal to Given Figure Less a Parallelogram
Proposition $29$: Construction of Parallelogram Equal to Given Figure Exceeding a Parallelogram
Proposition $30$: Construction of Golden Section
Proposition $31$: Similar Figures on Sides of Right-Angled Triangle
Proposition $32$: Triangles with Two Sides Parallel and Equal
Proposition $33$: Angles in Circles have Same Ratio as Arcs