Abariwo

Tangencies
Tangency is the state of being tangent; having contact at a single point or along a line
without crossing
A tangent [tændʒənt] is a geometric line, curve, plane, or curved surface that touches
another curve or surface at one point but does not intersect it

Tangent lines
In geometry, the tangent line (or simply the tangent) to a plane curve at a given
point is the straight linethat "just touches" the curve at that point. More precisely, a
straight line is said to be a tangent of a curve y = f(x) at a point x = c on the curve if the
line passes through the point (c, f(c)) on the curve and has slope f'(c) where f' is the
derivative of f. A similar definition applies to space curves and curves in n-dimensional
Euclidean space.

As it passes through the point wherethe tangent line and the curve meet, or the
point of tangency, the tangent line is "going in the same direction" as the curve, and in
this sense it is the best straight-line approximation to the curve at that point.

Tangent circles

Two pairs of tangent circles. Above internally and below externally tangent

Two circles of non-equal radius, both in the same plane, are said to be tangent toeach
other if they meet at only one point. Equivalently, two circles, with radii of ri and
centres at (xi , yi), for i = 1, 2 are said to be tangent to each other if
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Two circles are externally tangent if the distance between their centres is equal
to the sum of their radii.

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Two circles are internally tangent if the distance between their centres is equal
to the differencebetween their radii.

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Tangent planes
Similarly, the tangent plane to a surface at a given point is the plane that "just
touches" the surface at that point. The concept of a tangent is one of the most
fundamental notions in differential geometry and has been extensively generalized; see
Tangent space.

The word "tangent" comes from the Latin tangens, meaning "touching".

Tangentlines to circles
In Euclidean plane geometry, tangent lines to circles form the subject of several
theorems, and play an important role in many geometrical constructions and proofs.
Since the tangent line to a circle at a point P is perpendicular to the radius to that point,
theorems involving tangent lines often involve radial lines and orthogonal circles.

Tangent lines to one circle
Atangent line t to a circle C intersects the circle at a single point T. For
comparison, secant lines intersect a circle at two points, whereas another line may not
intersect a circle at all. This property of tangent lines is preserved under many
geometrical transformations, such as scalings, rotations, translations, inversions, and
map projections. In technical language, these transformations do notchange the
incidence structure of the tangent line and circle, even though the line and circle may be
deformed.

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The radius of a circle is perpendicular to the tangent line through its endpoint on
the circle's circumference. Conversely, the perpendicular to a radius through the same
endpoint is a tangent line. The resulting geometrical figure of circle and tangent line has
areflection symmetry about the axis of the radius.

By the power-of-a-point theorem, the product of lengths PM·PN for any ray PMN equals to the
square of PT, the length of the tangent line segment (red).

No tangent line can be drawn through a point in the interior of a circle, since any
such line must be a secant line. However, two tangent lines can be drawn to a circle
from a point P outside ofthe circle. The geometrical figure of circle and both tangent
lines likewise have a reflection symmetry about the radial axis joining P to the centre
point O of the circle. Thus length 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...