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Thus the lengths of the segments from P to the two tangent points are equal. By the secant-tangent theorem, the square of this tangent length equals the power of the point P in the circle C. This power equals the product of distances from P to any two intersection points of the circle with a secant line passing 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.
If a tangent from an external point A meets the circle at F and a secant from the external point A meets the circle at C and D respectively, then AF 2 = AC × AD (tangent–secant theorem). The angle between a chord and the tangent at one of its endpoints is equal to one half the angle subtended at the centre of the circle, on the opposite side ...
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 subtends the same arc on the circle.
The theorem does not apply to systems of circles with more than two circles tangent to each other at the same point. It requires that the points of tangency be distinct. [8] When more than two circles are tangent at a single point, there can be infinitely many such circles, with arbitrary curvatures; see pencil of circles. [27]
A stronger form of the circle packing theorem asserts that any polyhedral graph and its dual graph can be represented by two circle packings, such that the two tangent circles representing a primal graph edge and the two tangent circles representing the dual of the same edge always have their tangencies at right angles to each other at the same ...
Thales’ theorem: if AC is a diameter and B is a point on the diameter's circle, the angle ∠ ABC is a right angle. In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line AC is a diameter, the angle ∠ ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and ...
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 ...