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Let O 1 and O 2 be the centers of the two circles, C 1 and C 2 and let r 1 and r 2 be their radii, with r 1 > r 2; in other words, circle C 1 is defined as the larger of the two circles. Two different methods may be used to construct the external and internal tangent lines.
In the special cases of one of the diagonals or sides being a diameter of the circle, this theorem gives rise directly to the angle sum and difference trigonometric identities. [17] The relationship follows most easily when the circle is constructed to have a diameter of length one, as shown here.
Angle AOB is a central angle. A central angle is an angle whose apex (vertex) is the center O of a circle and whose legs (sides) are radii intersecting the circle in two distinct points A and B. Central angles are subtended by an arc between those two points, and the arc length is the central angle of a circle of radius one (measured in radians). [1]
The cosine rule may be used to give the angles A, B, and C but, to avoid ambiguities, the half angle formulae are preferred. Case 2: two sides and an included angle given (SAS). The cosine rule gives a and then we are back to Case 1. Case 3: two sides and an opposite angle given (SSA). The sine rule gives C and then we have Case 7. There are ...
If the angle subtended by the chord at the centre is 90°, then ℓ = r √2, where ℓ is the length 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 (⌢ and ⌢).
Creating the one point or two points in the intersection of two circles (if they intersect). For example, starting with just two distinct points, we can create a line or either of two circles (in turn, using each point as centre and passing through the other point). If we draw both circles, two new points are created at their intersections.
For example, one sphere that is described in Cartesian coordinates with the equation x 2 + y 2 + z 2 = c 2 can be described in spherical coordinates by the simple equation r = c. (In this system— shown here in the mathematics convention —the sphere is adapted as a unit sphere , where the radius is set to unity and then can generally be ...
[2] An inversion centered at p transforms A and B into concentric circles. [3] The midpoint of the two limiting points is the point where the radical axis of A and B crosses the line through their centers. This intersection point has equal power distance to all the circles in the pencil containing A and B. The limiting points themselves can be ...