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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.
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. External tangents Construction of the outer tangent
In geometry, the limiting points of two disjoint circles A and B in the Euclidean plane are points p that may be defined by any of the following equivalent properties: The pencil of circles defined by A and B contains a degenerate (radius zero) circle centered at p. [1] Every circle or line that is perpendicular to both A and B passes through p ...
In trigonometry, the law of tangents or tangent rule [1] is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides. In Figure 1, a , b , and c are the lengths of the three sides of the triangle, and α , β , and γ are the angles opposite those three respective sides.
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.
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For fixed points A and B, the set of points M in the plane, for which the angle ∠AMB is equal to α, is an arc of a circle. The measure of ∠AOB, where O is the center of the circle, is 2α. The inscribed angle theorem states that an angle θ inscribed in a circle is half of the central angle 2θ that intercepts the same arc on the circle
Inversion in a circle centered on P transforms the two given circles into new circles, and the solution circle into a line. Therefore, the transformed solution is a line that is tangent to the two transformed given circles. There are four such solution lines, which may be constructed from the external and internal homothetic centers of the two ...