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equation of a circle
1. To find the equation of a circle, observe that each point on it forms a right-angled triangle whose sides are the x distance and y distance and whose hypotenuse is r. Pythagoras's theorem tells us that x 2 + y 2 = r 2. For points outside the circle x 2 + y 2 exceeds r 2; inside it x 2 + y 2 is less than r 2. Along the intersection of these two regions the equation balances. All mathematical curves divide space in this way.
terms associated with a circle
2. Terms associated with a circle
Angles subtended by the same arc are equal
3. Inscribed angles of a circle
A circle is the set of all points in a plane at a given distance r, called the radius, from a fixed point, called the center. A circle is a simple closed curve that divides the plane into an interior and exterior. It has a perimeter, called a circumference, of length 2πr, and encloses an area of πr 2.

In coordinate geometry a circle with center (x0, y0) and radius r is the set of all points (x, y) such that:

(x - x0) 2 + (y - y0) 2 = r  2.

"Circle" comes from the Latin circus, which refers to a large round or rounded oblong enclosure in which the famous Roman chariot races were held.

A line cutting a circle in two places is called a secant. The segment of a secant bound by the circle is called a chord, and the longest chord is that which passes through the center and is known as a diameter. The ratio of the circumference to the diameter is π. The length of a circle between two radii is called an arc; the ratio between the length of an arc and the radius defines the angle between two radii in radians. The area bounded by two radii and an arc is known as a sector.
A segment is part of a circle bounded by a chord and the arc subtending the chord. A line touching a circle in one place is called a tangent. Tangent lines are perpendicular to radii.

In affine geometry all circles and ellipses become congruent, and in projective geometry the other conic sections join them. A circle is a conic section with eccentricity zero. In topology all simple closed curves are homeomorphic to circles, and the word circle is often applied to them as a result. The three-dimensional analog of the circle is the sphere, and the four-dimensional analog is the hypersphere.

Inscribed angle theorems

Referring to the bottom diagram (3), angle O = 2B = 2E. The general theorem that follows from this is that an inscribed is half the central angle. Also, B = E. In general, angles subtended by the same arc are equal.

Note, by the way, that the circumference of the arc ADC is πrO/180.

Johnson's theorem

If three congruent circles all intersect in a single point, then the other three points of intersection will lie on another circle of the same radius. This simple little theorem was discovered by Roger Johnson in 1916.

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