The document provides information about circles including definitions, properties, theorems and history. It defines a circle as a simple closed curve where all points are equidistant from the center. Key properties discussed are that a circle's circumference and radius are proportional, and its area is proportional to the square of the radius. Theorems covered relate to chords, tangents, secants and inscribed angles. The document also discusses squaring the circle problem and circles in history from ancient Greeks to modern mathematics.

1. Math's Assignment
Research on Circle

2.  A circle is a simple closed shape in Euclidean geometry.
 It is the set of all points in a plane that are at a given distance from a given point, the center;
equivalently it is the curve traced out by a point that moves so that its distance from a given
point is constant.
 The distance between any of the points and the center is called the radius.
 A circle is a simple closed curve which divides the plane into two regions: an interior and an
exterior.
 In everyday use, the term "circle" may be used interchangeably to refer to either the boundary
of the figure, or to the whole figure including its interior; in strict technical usage, the circle is
only the boundary and the whole figure is called a disc.
 A circle may also be defined as a special kind of ellipse in which the two foci are coincident
and the eccentricity is 0, or the two-dimensional shape enclosing the most area per unit
perimeter squared, using calculus of variations.
 A circle is a plane figure bounded by one line, and such that all right lines drawn from a certain
point within it to the bounding line, are equal. The bounding line is called its circumference
and the point, its center.

3. Terminology
 Annulus: the ring-shaped object, the region bounded by two concentric circles.
 Arc: any connected part of the circle.
 Centre: the point equidistant from the points on the circle.
 Chord: a line segment whose endpoints lie on the circle.
 Circumference: the length of one circuit along the circle, or the distance around the
circle.
 Diameter: a line segment whose endpoints lie on the circle and which passes through
Centre; or the length of such a line segment, which is the largest distance between any
two points on the circle. It is a special case of a chord, namely the longest chord, and it
twice the radius.
 Disc: the region of the plane bounded by a circle
 Lens: the intersection of two discs

4.  Passant: a coplanar straight line that does not touch the circle.
 Radius: a line segment joining the Centre of the circle to any point on the circle itself; or
the length of such a segment, which is half a diameter.
 Sector: a region bounded by two radii and an arc lying between the radii.
 Segment: a region, not containing the Centre, bounded by a chord and an arc lying
between the chord's endpoints.
 Secant: an extended chord, a coplanar straight line cutting the circle at two points.
 Semicircle: an arc that extends from one of a diameter's endpoints to the other. In non-
technical common usage it may mean the diameter, arc, and its interior, a two
dimensional region, that is technically called a half-disc. A half-disc is a special case of
a segment, namely the largest one.
 Tangent: a coplanar straight line that touches the circle at a single point.

5. Chord, secant, tangent, radius, and
diameter, Arc, sector, and segment.

6. In History
 The word circle derives from the Greek κίρκος/κύκλος (kirkos/kuklos), itself
the Homeric Greek κρίκος (krikos), meaning "hoop" or "ring". The origins of the
words circus and circuit are closely related.
 The circle has been known since before the beginning of recorded history.
 Natural circles would have been observed, such as the Moon, Sun, and a short plant
blowing in the wind on sand, which forms a circle shape in the sand.
 The circle is the basis for the wheel, which, with related inventions such as gears, makes
much of modern machinery possible.
 In mathematics, the study of the circle has helped inspire the development of
geometry, astronomy, and calculus.

7.  Early science, particularly geometry and astrology and astronomy, was
connected to the divine for most medieval scholars, and many believed
that there was something intrinsically "divine" or "perfect" that could be
found in circles.
 The compass in this 13th-century manuscript is a symbol of God's act
of Creation. Notice also the circular shape of the halo.
 1700 BCE – The Rhind papyrus gives a method to find the area of a
circular field. The result corresponds to 256/81 (3.16049...) as an
approximate value of π. 300 BCE – Book 3 of Euclid's Elements deals with
the properties of circles.
 In Plato's Seventh Letter there is a detailed definition and explanation of
the circle. Plato explains the perfect circle, and how it is different from any
drawing, words, definition or explanation.
 1880 CE – Lindemann proves that π is transcendental, effectively settling
the millennia-old problem of squaring the circle.

8. Length of Circumference
 The ratio of a circle's circumference to its diameter is π (pi), an irrational constant approximately
equal to 3.141592654.
 Thus the length of the circumference C is related to the radius r and
diameter d by:
 Area enclosed:
 Area enclosed by a circle = π × area of the shaded square
 As proved by Archimedes, in his Measurement of a Circle, the area enclosed by a circle is equal to
that of a triangle whose base has the length of the circle's circumference and whose height equals
the circle's radius,[7] which comes to π multiplied by the radius squared:
 Equivalently, denoting diameter by d, that is, approximately 79% of the circumscribing square
(whose side is of length d).The circle is the plane curve enclosing the maximum area for a given arc
length. This relates the circle to a problem in the calculus of variations, namely the isoperimetric
inequality.

9. Properties
 The circle is the shape with the largest area for a given length of perimeter.
 The circle is a highly symmetric shape: every line through the Centre forms a line of reflection
symmetry and it has rotational symmetry around the Centre for every angle. Its symmetry group is
the orthogonal group O(2,R). The group of rotations alone is the circle group T.
 All circles are similar.
 A circle's circumference and radius are proportional.
 The area enclosed and the square of its radius are proportional.
 The constants of proportionality are 2π and π, respectively.
 The circle which is centered at the origin with radius 1 is called the unit circle.
 Thought of as a great circle of the unit sphere, it becomes the Riemannian circle.
 Through any three points, not all on the same line, there lies a unique circle. In Cartesian
coordinates, it is possible to give explicit formulae for the coordinates of the Centre of the circle
and the radius in terms of the coordinates of the three given points.

10. Chord
 Chords are equidistant from the Centre of a circle if and only if they are equal in length.
 The perpendicular bisector of a chord passes through the Centre of a circle; equivalent
stemming from the uniqueness of the perpendicular bisector are:
 A perpendicular line from the Centre of a circle bisects the chord.
 The line segment through the Centre bisecting a chord is perpendicular to the chord.
 If a central angle and an inscribed angle of a circle are subtended by the same chord and on
same side of the chord, then the central angle is twice the inscribed angle.
 If two angles are inscribed on the same chord and on the same side of the chord, then they are
equal.
 If two angles are inscribed on the same chord and on opposite sides of the chord, then they
are supplementary.
 For a cyclic quadrilateral, the exterior angle is equal to the interior opposite angle.
 An inscribed angle subtended by a diameter is a right angle.

11.  The diameter is the longest chord of the circle.
 If the intersection of any two chords divides one chord into lengths a and b and divides the
other chord into lengths c and d, then ab = cd.
 If the intersection of any two perpendicular chords divides one chord into lengths a and b and
divides the other chord into lengths c and d, then a2 + b2 + c2 + d2 equals the square of the
diameter.
 The sum of the squared lengths of any two chords intersecting at right angles at a given point
is the same as that of any other two perpendicular chords intersecting at the same point, and
is given by 8r 2 – 4p 2 (where r is the circle's radius and p is the distance from the Centre point
to the point of intersection).
 The distance from a point on the circle to a given chord times the diameter of the circle equals
the product of the distances from the point to the ends of the chord.

12. Tangent
 A line drawn perpendicular to a radius through the end point of the radius lying
the circle is a tangent to the circle.
 A line drawn perpendicular to a tangent through the point of contact with a circle
passes through the Centre of the circle.
 Two tangents can always be drawn to a circle from any point outside the circle,
these tangents are equal in length.
 If a tangent at A and a tangent at B intersect at the exterior point P, then denoting
the Centre as O, the angles ∠BOA and ∠BPA are supplementary.
 If AD is tangent to the circle at A and if AQ is a chord of the circle,
then ∠DAQ = 2arc(AQ).

13. Theorems
 The chord theorem states that if two chords, CD and EB,
intersect at A, then CA × DA = EA × BA.
 If a tangent from an external point D meets the circle at C and
a secant from the external point D meets the circle at G and E respectively, then DC2 = DG × DE.
 If two secants, DG and DE, also cut the circle at H and F respectively, then DH × DG = DF × DE.
 The angle between a tangent and chord is equal to one half the angle subtended at the Centre
the circle, on the opposite side of the chord.
 If the angle subtended by the chord at the Centre is 90 degrees then l = r √2, where l is the
of the chord and r is the radius of the circle.
 If two secants are inscribed in the circle as shown at right, then the measurement of angle A is
equal to one half the difference of the measurements of the enclosed arcs (DE and BC). This is
secant-secant theorem.

14. Inscribed Angles
 An inscribed angle (examples are the blue and green angles in the figure) is exactly
the corresponding central angle (red).
 Hence, all inscribed angles that subtend the same arc (pink) are equal.
 Angles inscribed on the arc (brown) are supplementary. In particular, every inscribed
angle that subtends a diameter is a right angle (since the central angle is 180

15. Sagitta
 The sagitta (also known as the versine) is a line segment drawn perpendicular to a chord,
the midpoint of that chord and the arc of the circle.
 Given the length y of a chord, and the length x of the sagitta, the Pythagorean theorem can be
used to calculate the radius of the unique circle which will fit around the two lines:
 Another proof of this result which relies only on two chord properties given above is as follows.
 Given a chord of length y and with sagitta of length x, since the sagitta intersects the midpoint
the chord, we know it is part of a diameter of the circle.
 Since the diameter is twice the radius, the "missing" part of the diameter is (2r − x) in length.
 Using the fact that one part of one chord times the other part is equal to the same product
along a chord intersecting the first chord, we find that (2r − x)x = (y / 2)2. Solving for r, we find
required result.

16. Circle of Apollonius
 Apollonius of Perga showed that a circle may also be defined as the set of points in a
plane having a constant ratio (other than 1) of distances to two fixed
set of points where the distances are equal is the perpendicular bisector of A and B,
line.)
 That circle is sometimes said to be drawn about two points.
 The proof is in two parts. First, one must prove that, given two foci A and B and a
of distances, any point P satisfying the ratio of distances must fall on a particular
 Let C be another point, also satisfying the ratio and lying on segment AB. By
bisector theorem the line segment PC will bisect the interior angle APB, since the
segments are similar

17.  Analogously, a line segment PD through some point D on AB extended bisects the
corresponding exterior angle BPQ where Q is on AP extended.
 Since the interior and exterior angles sum to 180 degrees, the angle CPD is exactly 90
degrees, i.e., a right angle. The set of points P such that angle CPD is a right angle forms a
circle, of which CD is a diameter.
 Second for a proof that every point on the indicated circle satisfies the given ratio.

18. Cross Ratio
 A closely related property of circles involves the geometry of the cross-ratio of points in
the complex plane.
 If A, B, and C are as above, then the circle of Apollonius for these three points is the
collection of points P for which the absolute value of the cross-ratio is equal to one:
 Stated another way, P is a point on the circle of Apollonius if and only if the cross-ratio
[A,B;C,P] is on the unit circle in the complex plane.

19. Generalised circles
 If C is the midpoint of the segment AB, then the collection of
points P satisfying the Apollonius condition is not a circle, but rather a line.
 Thus, if A, B, and C are given distinct points in the plane, then the locus of
points P satisfying the above equation is called a "generalised circle." It may
either be a true circle or a line. In this sense a line is a generalised circle of
infinite radius.

20. Circle as limiting case of other figures
The circle can be viewed as a limiting case of each of various other figures:
 A Cartesian oval is a set of points such that a weighted sum of the distances from any of its
points to two fixed points (foci) is a constant.
 An ellipse is the case in which the weights are equal.
 A circle is an ellipse with an eccentricity of zero, meaning that the two foci coincide with each
other as the centre of the circle.
 A circle is also a different special case of a Cartesian oval in which one of the weights is zero.
 A superellipse has an equation of the form for positive a, b, and n.
 A supercircle has b = a. A circle is the special case of a supercircle in which n = 2.
 A Cassini oval is a set of points such that the product of the distances from any of its points to
two fixed points is a constant. When the two fixed points coincide, a circle results.

21. Squaring the circle
 Squaring the circle is the problem, proposed by ancient geometers, of constructing a square
with the same area as a given circle by using only a finite number of steps withcompass and
straightedge.
 In 1882, the task was proven to be impossible, as a consequence of the Lindemann–
theorem which proves that pi (π) is a transcendental number, rather than analgebraic irrational
number; that is, it is not the root of any polynomial with rational coefficients.
 Interactive Java applets for the properties of and elementary constructions involving circles.
 Interactive Standard Form Equation of Circle Click and drag points to see standard form
equation in action
 Munching on Circles at cut-the-knot

22. Presented by
R.Sanjith Varun IX-B