# Book:Euclid/The Elements/Book III

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## Euclid:

## Euclid: *The Elements: Book III*

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

### Contents

- Proposition $1$: Finding Center of Circle
- Proposition $2$: Chord Lies Inside its Circle
- Proposition $3$: Conditions for Diameter to be Perpendicular Bisector
- Proposition $4$: Chords do not Bisect Each Other
- Proposition $5$: Intersecting Circles have Different Centers
- Proposition $6$: Touching Circles have Different Centers
- Proposition $7$: Relative Lengths of Lines Inside Circle
- Proposition $8$: Relative Lengths of Lines Outside Circle
- Proposition $9$: Condition for Point to be Center of Circle
- Proposition $10$: Two Circles have at most Two Points of Intersection
- Proposition $11$: Line Joining Centers of Two Circles Touching Internally
- Proposition $12$: Line Joining Centers of Two Circles Touching Externally
- Proposition $13$: Circles Touch at One Point at Most
- Proposition $14$: Equal Chords in Circle
- Proposition $15$: Relative Lengths of Chords of Circlesâ€Ž
- Proposition $16$: Line at Right Angles to Diameter of Circle
- Proposition $17$: Construction of Tangent from Point to Circle
- Proposition $18$: Radius at Right Angle to Tangent
- Proposition $19$: Right Angle to Tangent of Circle goes through Center
- Proposition $20$: Inscribed Angle Theorem
- Proposition $21$: Angles in Same Segment of Circle are Equal
- Proposition $22$: Opposite Angles of Cyclic Quadrilateral sum to Two Right Angles
- Proposition $23$: Segment on Given Base Unique
- Proposition $24$: Similar Segments on Equal Bases are Equal
- Proposition $25$: Construction of Circle from Segment
- Proposition $26$: Equal Angles in Equal Circles
- Proposition $27$: Angles on Equal Arcs are Equal
- Proposition $28$: Straight Lines Cut Off Equal Arcs in Equal Circles
- Proposition $29$: Equal Arcs of Circles Subtended by Equal Straight Lines
- Proposition $30$: Bisection of Arc
- Proposition $31$: Relative Sizes of Angles in Segments
- Proposition $32$: Tangent-Chord Theorem
- Proposition $33$: Construction of Segment on Given Line Admitting Given Angle
- Proposition $34$: Construction of Segment on Given Circle Admitting Given Angle
- Proposition $35$: Intersecting Chords Theorem
- Proposition $36$: Tangent Secant Theorem
- Proposition $37$: Converse of Tangent Secant Theorem