# Category:Circles

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This category contains results about Circles.

Definitions specific to this category can be found in Definitions/Circles.

In the words of Euclid:

*A***circle**is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another;

(*The Elements*: Book $\text{I}$: Definition $15$)

## Subcategories

This category has the following 18 subcategories, out of 18 total.

### A

### C

### E

### F

### I

### R

### S

### T

## Pages in category "Circles"

The following 95 pages are in this category, out of 95 total.

### A

- Angles in Circles have Same Ratio as Arcs
- Angles in Same Segment of Circle are Equal
- Angles made by Chord with Tangent
- Angles on Equal Arcs are Equal
- Arc Length of Sector
- Area between Two Non-Intersecting Chords
- Area of Circle
- Area of Sector
- Area of Segment of Circle
- Areas of Circles are as Squares on Diameters
- Areas of Circles are as Squares on Diameters/Lemma
- Areas of Similar Polygons Inscribed in Circles are as Squares on Diameters

### B

### C

- Chord Lies Inside its Circle
- Chords do not Bisect Each Other
- Circle is Bisected by Diameter
- Circles Touch at One Point at Most
- Circumscribing about Circle Triangle Equiangular with Given
- Circumscribing Circle about Regular Pentagon
- Circumscribing Circle about Square
- Circumscribing Circle about Triangle
- Circumscribing Regular Pentagon about Circle
- Circumscribing Square about Circle
- Concentric Circles do not Intersect
- Condition for Point to be Center of Circle
- Conditions for Diameter to be Perpendicular Bisector
- Construction of Circle from Segment
- Construction of Equilateral Polygon with Even Number of Sides in Outer of Concentric Circles
- Construction of Isosceles Triangle whose Base Angle is Twice Apex
- Construction of Segment on Given Circle Admitting Given Angle
- Construction of Segment on Given Line Admitting Given Angle
- Construction of Tangent from Point to Circle
- Converse of Tangent Secant Theorem

### E

### I

- Inscribed Angle Theorem
- Inscribing Circle in Regular Pentagon
- Inscribing Circle in Square
- Inscribing Circle in Triangle
- Inscribing in Circle Triangle Equiangular with Given
- Inscribing Regular 15-gon in Circle
- Inscribing Regular 15-gon in Circle/Corollary
- Inscribing Regular Hexagon in Circle
- Inscribing Regular Hexagon in Circle/Porism
- Inscribing Regular Pentagon in Circle
- Inscribing Square in Circle
- Intersecting Chord Theorem
- Intersecting Circles have Different Centers

### L

### P

### R

- Radius at Right Angle to Tangent
- Ratio of Chord to Arc
- Ratios of Sizes of Mutually Inscribed Multidimensional Cubes and Spheres
- Regiomontanus' Angle Maximization Problem
- Relative Lengths of Chords of Circles
- Relative Lengths of Lines Inside Circle
- Relative Lengths of Lines Outside Circle
- Relative Sizes of Angles in Segments
- Right Angle to Tangent of Circle goes through Center
- Round Peg fits in Square Hole better than Square Peg fits in Round Hole

### S

- Secant Secant Theorem
- Segment on Given Base Unique
- Similar Segments on Equal Bases are Equal
- Six Equal Circles Tangent to Equal Circle
- Square Inscribed in Circle is greater than Half Area of Circle
- Square on Side of Equilateral Triangle inscribed in Circle is Triple Square on Radius of Circle
- Squaring the Circle
- Straight Lines Cut Off Equal Arcs in Equal Circles