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In Euclidean plane geometry, a tangent line to a circle is a line that touches the circle at exactly one point, never entering the circle's interior. Tangent lines to circles form the subject of several theorems , and play an important role in many geometrical constructions and proofs .
Tangent to a curve. The red line is tangential to the curve at the point marked by a red dot. Tangent plane to a sphere. In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point.
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 ...
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 ...
All tangent circles to the given circles can be found by varying line . Positions of the centers Circles tangent to two circles. If is the center and the radius of the circle, that is tangent to the given circles at the points ,, then:
By definition, a point is tangent to a circle or a line if it intersects them, that is, if it lies on them; thus, two distinct points cannot be tangent. If the angle between lines or circles at an intersection point is zero, they are said to be tangent; the intersection point is called a tangent point or a point of tangency.
The point of osculation is also called the double cusp. Contact is a geometric notion; it can be defined algebraically as a valuation. One speaks also of curves and geometric objects having k-th order contact at a point: this is also called osculation (i.e. kissing), generalising the property of being tangent. (Here the derivatives are ...
If two tangent lines can be drawn from a pole to the circle, then its polar passes through both tangent points. If a point lies on the circle, its polar is the tangent through this point. If a point P lies on its own polar line, then P is on the circle. Each line has exactly one pole.
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