# Section: Book 04: Circles: Inscription and Circumscription

The 4th Book of the "Elements" is devoted to the solution of problems, chiefly relating to the inscription and circumscription of regular polygons and circles. It contains 16 propositions proven purely geometrically.

The focus of this book lies on compass and ruler constructions. It contains the following constructions:

• Inscribing a chord with a given length into a given circle (Prop. 4.01),
• Inscribing a triangle with given angles into a given circle (Prop. 4.02).
• Circumscribing a triangle with given angles about a given circle (Prop. 4.03).
• Inscribing a circle into a given triangle (Prop. 4.04).
• Circumscribing a circle about a given triangle (Prop. 4.05).
• Inscribing a square into a given circle (Prop. 4.06).
• Circumscribing a square about a given circle (Prop. 4.07).
• Inscribing a circle into a given square (Prop. 4.08).
• Circumscribing a circle about a given square (Prop. 4.09).
• Constructing an isosceles triangle with the angles $72^\circ$, $72^\circ$ and $36^\circ$ (Prop. 4.10).
• Inscribing a regular pentagon into a given circle (Prop. 4.11).
• Circumscribing a regular pentagon about a given circle (Prop. 4.12).
• Inscribing a circle into a given regular pentagon (Prop. 4.13).
• Circumscribing a circle about a given regular pentagon (Prop. 4.14).
• Inscribing a regular hexagon into a given circle (Prop. 4.15).
• Circumscribing a regular hexagon about a given circle (Corollary to Prop. 4.15).
• Inscribing a regular pentakaidecagon (15-gon) into a given circle (Prop. 4.16).

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### References

#### Bibliography

1. Byrne, O.: "The First Six Books of the Elements of Euclid, in Which Coloured Diagrams and Symbols are used Instead of Letters", London William Pickering, 1847