Questions and Answers
What is the equation of a circle in polar coordinates?
r = a
What is the relationship between an inscribed angle and its corresponding central angle?
An inscribed angle is half the corresponding central angle.
What is the circle of Apollonius for three points?
The collection of points P for which the absolute value of the crossratio is equal to one.
What is a hypocycloid?
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What did the LindemannWeierstrass theorem prove about the circlesquaring problem?
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What is the significance of the circle in mystical doctrines?
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What is a tangential polygon?
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What is the definition of a circle?
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What is the ratio of a circle's circumference to its diameter?
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Study Notes
Circle in Euclidean Geometry

A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre.

A circle is a simple closed curve that divides the plane into two regions: an interior and an exterior.

The term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior.

The circle has been known since before the beginning of recorded history.

The ratio of a circle's circumference to its diameter is π (pi), an irrational constant approximately equal to 3.141592654.

As proved by Archimedes, 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.

The circle is the plane curve enclosing the maximum area for a given arc length.

In an x–y Cartesian coordinate system, the circle with centre coordinates (a, b) and radius r is the set of all points (x, y) such that (xa)^2 + (yb)^2 = r^2.

In polar coordinates, the equation of a circle is r = a, where a is the radius of the circle.

In the complex plane, a circle with a centre at c and radius r has the equation zc^2 = r^2.

The word circle derives from the Greek κίρκος/κύκλος (kirkos/kuklos), meaning "hoop" or "ring".

The study of the circle has helped inspire the development of geometry, astronomy and calculus.Properties, Theorems, and Constructions of Circles

A generalised circle is either a circle or a line.

The tangent line through a point P on the circle is perpendicular to the diameter passing through P.

An inscribed angle is half the corresponding central angle, and all inscribed angles that subtend the same arc are equal.

The sagitta is a line segment drawn perpendicular to a chord, between the midpoint of that chord and the arc of the circle.

There are many compassandstraightedge constructions resulting in circles.

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 foci, A and B.

The circle of Apollonius for three points is the collection of points P for which the absolute value of the crossratio is equal to one.

A tangential polygon is any convex polygon within which a circle can be inscribed that is tangent to each side of the polygon.

A cyclic polygon is any convex polygon about which a circle can be circumscribed, passing through each vertex.

A hypocycloid is a curve that is inscribed in a given circle by tracing a fixed point on a smaller circle that rolls within and tangent to the given circle.

A circle can be viewed as a limiting case of various other figures, such as squares under taxicab geometry.

The locus of points such that the sum of the squares of the distances to a finite set of points in the plane is constant is a circle, whose centre is at the centroid of the given points.The CircleSquaring Problem and Its Impossibility

The problem involves constructing a square with the same area as a given circle using only a finite number of steps with compass and straightedge.

In 1882, the Lindemann–Weierstrass theorem proved that the task is impossible due to pi (π) being a transcendental number, rather than an algebraic irrational number.

Despite the problem's impossibility, it remains of interest for pseudomath enthusiasts.

The circle has been used in visual art since the earliest known civilizations to convey messages and express ideas.

Differences in worldview impacted how artists perceived the circle, with some emphasizing its perimeter to demonstrate democratic manifestation and others focusing on its center to symbolize cosmic unity.

The circle symbolizes many sacred and spiritual concepts, including unity, infinity, wholeness, the universe, divinity, balance, stability, and perfection.

The circle has been conveyed in cultures worldwide through symbols such as a compass, halo, vesica piscis, ouroboros, Dharma wheel, rainbow, mandalas, and rose windows.

The circle is a significant symbol in mystical doctrines, representing the infinite and cyclical nature of existence.

In religious traditions, the circle represents heavenly bodies and divine spirits.

The circle's significance has endured throughout history and continues to inspire artists and thinkers today.

The circle has been used in various forms of art, including painting, sculpture, architecture, and design.

Studying the circle's symbolism provides insight into the beliefs and values of different cultures and civilizations.
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Description
Test your knowledge of the circle in Euclidean geometry with this informative quiz! From its definition as a shape consisting of points equidistant from a center to its properties, theorems, and constructions, this quiz covers a wide range of topics related to the circle. You'll also learn about the circlesquaring problem and its impossibility, as well as the circle's significance in art, culture, and spirituality. Whether you're a math enthusiast or simply curious about the circle's role in