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Tangent To A Circle

Check us out at A tangent 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, rotation, translations, inversions, and map projections. In technical language, these transformations do not change the incidence structure of the tangent line and circle, even though the line and circle may be deformed. 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 a reflection 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 of the circle. The geometrical figure of circle and both tangent lines likewise have a reflection symmetry about the radial axis joining P to the center 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 distances from P to any two intersection points of the circle with a secant line passing through P. The angle PTO between a tangent line and a radial line segment is a right angle. By the inscribed angle theorem, the angle PTS between the tangent line and another point on the circumference equals half of the central angle TOS subtended by the two points T and S. The tangent line t and the tangent point T have a conjugate relationship to one another, which has been generalized into the idea of pole points and polar lines. The same reciprocal relation exists between a point P outside the circle and the secant line joining its two points of tangency.
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