This subsection contains the propositions from Book 4 of Euclid's “Elements”.

- Proposition: 4.01: Fitting Chord Into Circle
- Proposition: 4.02: Inscribing in Circle Triangle Equiangular with Given Angles
- Proposition: 4.03: Circumscribing about Circle Triangle Equiangular with Given Angles
- Proposition: 4.04: Inscribing Circle in Triangle
- Proposition: 4.05: Circumscribing Circle about Triangle
- Proposition: 4.06: Inscribing Square in Circle
- Proposition: 4.07: Circumscribing Square about Circle
- Proposition: 4.08: Inscribing Circle in Square
- Proposition: 4.09: Circumscribing Circle about Square
- Proposition: 4.10: Construction of Isosceles Triangle whose Base Angle is Twice Apex
- Proposition: 4.11: Inscribing Regular Pentagon in Circle
- Proposition: 4.12: Circumscribing Regular Pentagon about Circle
- Proposition: 4.13: Inscribing Circle in Regular Pentagon
- Proposition: 4.14: Circumscribing Circle about Regular Pentagon
- Proposition: 4.15: Side of Hexagon Inscribed in a Circle Equals the Radius of that Circle
- Proposition: 4.16: Inscribing Regular Pentakaidecagon in Circle