# Book:Euclid/The Elements/Book VI

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## Euclid:

## Euclid: *The Elements: Book VI*

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

### Contents

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

- 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
- 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
- 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