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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 ...
The value of the two products in the chord theorem depends only on the distance of the intersection point S from the circle's center and is called the absolute value of the power of S; more precisely, it can be stated that: | | | | = | | | | = where r is the radius of the circle, and d is the distance between the center of the circle and the ...
An Apollonian gasket with integer curvatures, generated by four mutually tangent circles with curvatures −10 (the outer circle), 18, 23, and 27. When four tangent circles described by equation (2) all have integer curvatures, the alternative fourth circle described by the second solution to the equation must also have an integer curvature ...
In geometry, tangent circles (also known as kissing circles) are circles in a common plane that intersect in a single point. There are two types of tangency : internal and external. Many problems and constructions in geometry are related to tangent circles; such problems often have real-life applications such as trilateration and maximizing the ...
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 similarity yields an equation for ratios which is equivalent to the equation of the theorem given above: = | | | | = | | | | Next to the intersecting chords theorem and the tangent-secant theorem , the intersecting secants theorem represents one of the three basic cases of a more general theorem about two intersecting lines and a circle ...
Trigonometric ratios can also be represented using the unit circle, which is the circle of radius 1 centered at the origin in the plane. [37] In this setting, the terminal side of an angle A placed in standard position will intersect the unit circle in a point (x,y), where x = cos A {\displaystyle x=\cos A} and y = sin A {\displaystyle ...
where C is the circumference of a circle, d is the diameter, and r is the radius. More generally, = where L and w are, respectively, the perimeter and the width of any curve of constant width. = where A is the area of a circle. More generally, =