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The equation of the circle determined by three points (,), (,), (,) not on a line is obtained by a conversion of the 3-point form of a circle equation: () + () () () = () + () () (). Homogeneous form In homogeneous coordinates , each conic section with the equation of a circle has the form x 2 + y 2 − 2 a x z − 2 b y z + c z 2 = 0 ...
The defining property of the Carlyle circle can be established thus: the equation of the circle having the line segment AB as diameter is x(x − s) + (y − 1)(y − p) = 0. The abscissas of the points where the circle intersects the x-axis are the roots of the equation (obtained by setting y = 0 in the equation of the circle)
Ptolemy used a circle of diameter 120, and gave chord lengths accurate to two sexagesimal (base sixty) digits after the integer part. [2] The chord function is defined geometrically as shown in the picture. The chord of an angle is the length of the chord between two points on a unit circle separated by that central angle.
Cartesian coordinate system with a circle of radius 2 centered at the origin marked in red. The equation of a circle is (x − a) 2 + (y − b) 2 = r 2 where a and b are the coordinates of the center (a, b) and r is the radius. In Cartesian geometry, equations are used to describe geometric figures.
Given three points A, B and C on a circle with center O, the angle ∠ AOC is twice as large as the angle ∠ ABC. A related result to Thales's theorem is the following: If AC is a diameter of a circle, then: If B is inside the circle, then ∠ ABC > 90° If B is on the circle, then ∠ ABC = 90° If B is outside the circle, then ∠ ABC < 90°.
Given a circle whose center is point O, choose three points V, C, D on the circle. Draw lines VC and VD: angle ∠DVC is an inscribed angle. Now draw line OV and extend it past point O so that it intersects the circle at point E. Angle ∠DVC intercepts arc DC on the circle. Suppose this arc includes point E within it.
A circle is the set of points in a plane that lie at radius from a center point . (,) = {:}In the complex plane, is a complex number and is a set of complex numbers. Using the property that a complex number multiplied by its conjugate is the square of its modulus (its Euclidean distance from the origin), an implicit equation for is:
If = + is the distance from c 1 to c 2 we can normalize by =, =, = to simplify equation (1), resulting in the following system of equations: + =, + =; solve these to get two solutions (k = ±1) for the two external tangent lines: = = + = (+) Geometrically this corresponds to computing the angle formed by the tangent lines and the line of ...