# 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 47 subcategories, out of 47 total.

### A

- Annuli (2 P)
- Area of Sector (3 P)

### B

- Broken Chord Theorem (6 P)

### C

- Circles Tangent to Y Axis (3 P)
- Circumcircles (empty)
- Circumference of Geometric Figure (empty)
- Concyclic Points (empty)

### E

- Eccentric Circles (empty)

### F

- Finding Center of Circle (4 P)
- Fontené Theorems (9 P)

### I

- Imaginary Circles (empty)
- Inscribe (empty)
- Inscribed Angle Theorem (3 P)
- Intersecting Chords Theorem (6 P)

### L

- Latus Rectum of Circle (empty)
- Length of Chord of Circle (3 P)

### O

- Orthogonal Circles (1 P)
- Osculating Circles (empty)

### P

- Point-Circles (empty)
- Power of a Point Theorem (1 P)

### S

- Secant Secant Theorem (4 P)
- Squaring the Circle (5 P)

### T

- Tangent Secant Theorem (4 P)
- Tangent-Chord Theorem (3 P)
- Tangents to Circles (9 P)
- Thales' Theorem (5 P)

## Pages in category "Circles"

The following 122 pages are in this category, out of 122 total.

### A

- Angle of Intersection of Circles equals Angle between Radii
- Angle of Intersection of Circles is Equal at Both Points
- Angles in Circles have Same Ratio as Arcs
- Angles in Same Segment of Circle are Equal
- Angles on Equal Arcs are Equal
- Arc Length of Sector
- Archimedes' Limits to Value of Pi
- 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

- Characteristic of Interior Point of Circle whose Center is Origin
- Characteristic of Quadratic Equation that Represents Circle
- Chord Lies Inside its Circle
- Chords do not Bisect Each Other
- Circle is Bisected by Diameter
- Circle is Curve of Second Degree
- Circle is Ellipse with Equal Major and Minor Axes
- Circles Touch at One Point at Most
- Circularity is not Invariant under Affine Transformation
- 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
- Condition for Circles to be Orthogonal
- Condition for Point to be Center of Circle
- Condition of Tangency to Circle whose Center is Origin
- Conditions for Diameter to be Perpendicular Bisector
- Construction of Circle from Segment
- 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

### E

- Eccentricity of Circle equals 0
- Equal Angles in Equal Circles
- Equal Arcs of Circles Subtended by Equal Straight Lines
- Equal Chords in Circle
- Equation of Chord of Contact on Circle Centered at Origin
- Equation of Circle
- Equation of Circle in Complex Plane
- Equation of Normal to Circle Centered at Origin
- Equation of Straight Line Tangent to Circle
- Equation of Tangent to Circle Centered at Origin
- Equation of Tangents to Circle from Point
- Evolute of Circle is its Center

### I

- Inscribe Square in Circle using Compasses Alone
- Inscribe Square in Circle using Compasses Alone/Corollary
- 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 Hexagon in Circle
- Inscribing Regular Hexagon in Circle/Porism
- Inscribing Regular Pentagon in Circle
- Inscribing Square in Circle
- Intersecting Chords Theorem
- Intersecting Circles have Different Centers

### L

- Length of Chord of Circle
- Length of Chord Projected from Point on Intersecting Circle
- Length of Tangent from Point to Circle center Origin
- Line at Right Angles to Diameter of Circle
- Line at Right Angles to Diameter of Circle/Porism
- Line Joining Centers of Two Circles Touching Externally
- Line Joining Centers of Two Circles Touching Internally

### O

### P

- Parallel Chords Cut Equal Chords in a Circle
- Parameterization of Unit Circle is Simple Loop
- Parametric Equation of Involute of Circle
- Perimeter of Circle
- Perpendicular Bisector of Chord Passes Through Center
- Perpendicular from Center of Circle to Side of Inscribed Pentagon
- Power of a Point Theorem
- Problem of Apollonius

### R

- 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
- 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
- Simple Loop Image Equals Set Homeomorphic to Circle
- 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

### T

- Tangent Secant Theorem
- Tangent-Chord Theorem
- Tangents to Circle from Point are of Equal Length
- Tangents to Circle from Point subtend Equal Angles at Center
- Tangents to Circle from Point subtend Equal Angles at Center/Corollary
- Thales' Theorem
- Thales' Theorem/Converse
- Three Points Describe a Circle
- Touching Circles have Different Centers
- Two Circles have at most Two Points of Intersection
- Two Non-Intersecting Circles have Four Common Tangents