User:XOR'easter/Circular reasoning

This is a workspace for material related to improving the article Circle, which is currently under-cited, heavy on bullet points, disorganized, etc.

A circle is a shape made of all the points in a plane that are the same distance from a given point, the centre. The distance from the centre to any point of the circle is the radius of the circle. The length of a line segment from a point on the circle, through the centre to the opposite side of the circle is the diameter. The distance around the circle is the circumference, and for any circle, the ratio of the circumference to the diameter is the number π (pi). A circle bounds a region of the plane called a disc, the area of which is π times the square of the radius.

Euclidean plane geometry

In Euclidean geometry, circles and straight lines are the basic shapes from which others are built. The two Euclidean tools for constructing shapes are the straightedge and the compass. The former is for drawing straight lines, and the latter is for drawing circles. One of the basic postulates of Euclid's Elements is that a circle can be drawn using any point as centre and with any radius.^[1]

Central and inscribed angles

Circles in triangle geometry

Given any triangle, there exists exactly one circle passing through all three of the triangle's vertices. This is known as the circumscribing circle, or circumcircle, of the triangle. Its center is the circumcenter of the triangle. Each triangle also has an inscribed circle, or incircle, which is the unique circle that lies within the circle and touches each of the three sides. Its center is the incenter of the triangle. The perpendicular bisector of a triangle side is the line that passes through the midpoint of that side at right angles; the three perpendicular bisectors of any triangle meet at its circumcenter.^[2] Likewise, the angle bisectors of a triangle are the three lines that split the angles at its vertices in half, and the three angle bisectors meet at the incenter. In any triangle, the midpoints of the three sides all lie on a common circle. Because this circle also passes through six other points of note, its is often called the nine-point circle of the triangle. The radius of the nine-point circle is half that of the circumcircle, and by Feuerbach's theorem, the nine-point circle is internally tangent to the incircle.

Cyclic quadrilaterals and polygons

A cyclic polygon is a polygon whose vertices all lie on a common circle, which is known as the circumscribing circle of that polygon. Every triangle is cyclic, as is every regular polygon.^[3] Not all quadrilaterals are cyclic, but for those that are, Ptolemy's theorem applies: the product of the lengths of two opposite sides, plus the product of the lengths of the two other opposite sides, equals the product of the lengths of the diagonals. Ptolemy's theorem reduces to the Pythagorean theorem when the quadrilateral is a rectangle. Because the sides of a cyclic quadrilateral are chords of a circle, and chords are related to trigonometric functions, Ptolemy's theorem is also one way to prove the addition formulae for sine and cosine.

Intersecting chords and secants

Pencils of circles

Squaring the circle

 

A circle and square with the same area

Conventionally, since ancient times, most units of area have been defined in terms of various squares, typically a square with a standard unit of length as its side, for example a square meter or square inch.^[4] In ancient Greek deductive geometry, the area of a planar shape was measured and compared by constructing a square with the same area by using only a finite number of steps with compass and straightedge, a process called quadrature or squaring. Euclid's Elements shows how to do this for rectangles, parallelograms, triangles, and then more generally for simple polygons by breaking them into triangular pieces.^[5] Some shapes with curved sides could also be squared, such as the lune of Hippocrates^[6] and the parabola.^[7] Because of this focus on quadrature as a measure of area, the Greeks and later mathematicians sought unsuccessfully to square the circle, constructing a square with the same area as a given circle, again using finitely many steps with a compass and straightedge. In 1882, the task was proven to be impossible as a consequence of the Lindemann–Weierstrass theorem. This theorem proves that pi (π) is a transcendental number rather than an algebraic irrational number; that is, it is not the root of any polynomial with rational coefficients. A construction for squaring the circle could be translated into a polynomial formula for π, which does not exist.^[8]

Characterizations and representations

Locus of points equidistant from a common center

Curve of constant curvature

Apollonian circles

Description in various coordinate systems

Intrinsic geometry of the circle

Circular motion and trigonometry

Circles in solid geometry

Circles as conic sections

Conic sections are curves made by the intersection of a plane with a cone. The type of curve produced depends upon the angle at which the plane passes through the cone. Different choices of angle produce hyperbolas, parabolas, and ellipses. A circle is a special case of ellipse, specifically, an ellipse with eccentricity equal to zero.

Non-Euclidean and inversive geometry

Representation using complex numbers

Oriented circles

Topological circles

Generalizations

Spheres of arbitrary dimension.

Curves of constant width?

L^p space

Related figures

Inscribed

Circumscribed

Curves found by transformation

Pedal, caustic, etc.

Applications

Geometry bookshelf

1. Byrne (2013), p. xxii.
2. Gonick (2024), pp. 87–88, 228.
3. Gonick (2024), pp. 228–229.
4. Treese, Steven A. (2018). "Historical Area". History and Measurement of the Base and Derived Units. Springer. pp. 301–390. doi:10.1007/978-3-319-77577-7_5. ISBN 978-3-319-77576-0.
5. Byrne (2013), p. 70.
6. Postnikov, M. M. (2000). "The problem of squarable lunes". American Mathematical Monthly. 107 (7): 645–651. doi:10.2307/2589121. JSTOR 2589121.
7. Berendonk, Stephan (2017). "Ways to square the parabola—a commented picture gallery". Mathematische Semesterberichte. 64 (1): 1–13. doi:10.1007/s00591-016-0173-0. MR 3629442.
8. Kasner, Edward (July 1933). "Squaring the circle". The Scientific Monthly. 37 (1): 67–71. JSTOR 15685.

• Abramson, Jay; et al. (2024). Algebra and Geometry 2e (digital ed.). OpenStax. ISBN 978-1-951693-40-4. Free textbook at the high-school level.
• Byrne, Oliver (2013) [1847]. The First Six Books of the Elements of Euclid (facsimile ed.). TASCHEN GmbH. ISBN 978-3-8365-4471-9.
• Cheng, Eugenia (2023). Is Math Real? How Simple Questions Lead Us To Mathematics' Deepest Truths. Basic Books. ISBN 978-1-541-6-01826. Some relevant material on how to think about sines and cosines.
• Coolidge, Julian (1916). A Treatise on the Circle and the Sphere. Oxford: Clarendon. (Internet Archive scan)
• Godfrey, Charles; Siddons, A. W. (1919). Elementary Geometry: Practical and Theoretical (3rd ed.). Cambridge University Press.
• Gonick, Larry (2024). The Cartoon Guide to Geometry. William Morrow. ISBN 978-0-06-315757-6. An axiomatic treatment of high-school geometry.
• Gonick, Larry (2011). The Cartoon Guide to Calculus. William Morrow. ISBN 978-0-06-168909-3. Called "an excellent treatment" by MAA Reviews [1].
• Maehara, Hiroshi; Martini, Horst (2024). Circles, Spheres and Spherical Geometry. Birkhäuser. doi:10.1007/978-3-031-62776-7. ISBN 978-3-031-62775-0.
• Moebs, William; et al. (2024). University Physics. Vol. 1 (digital ed.). OpenStax. ISBN 978-1-947172-20-3. Probably as good as any textbook for the basics of circular motion.
• O'Connor, John J.; Robertson, Edmund F., "Circle", MacTutor History of Mathematics Archive, University of St Andrews
• O'Connor, John J.; Robertson, Edmund F., "Squaring the circle", MacTutor History of Mathematics Archive, University of St Andrews
• Rich, Barnett (1963). Principles And Problems Of Plane Geometry. Schaum.
• Schorling, R.; Clark, John P.; Carter, H. W. (1935). Modern Mathematics: An Elementary Course. George G. Harrap & Co.
• Sidoli, Nathan; Thomas, Robert Spencer David, eds. (2023). The Spherics of Theodosios. London: Routledge. doi:10.4324/9781003142164. ISBN 0-367-55730-4.
• "Geometry and Trigonometry textbooks". Open Textbook Library. University of New Mexico.