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Common lines and line segments on a circle, including a secant. A straight line can intersect a circle at zero, one, or two points. A line with intersections at two points is called a secant line, at one point a tangent line and at no points an exterior line. A chord is the line segment that joins two distinct points of a circle. A chord is ...
The line segments OT 1 and OT 2 are radii of the circle C; since both are inscribed in a semicircle, they are perpendicular to the line segments PT 1 and PT 2, respectively. But only a tangent line is perpendicular to the radial line. Hence, the two lines from P and passing through T 1 and T 2 are tangent to the circle C.
The tangent-secant theorem can be proven using similar triangles (see graphic). Like the intersecting chords theorem and the intersecting secants theorem, the tangent-secant theorem represents one of the three basic cases of a more general theorem about two intersecting lines and a circle, namely, the power of point theorem.
In Euclidean geometry, the intersecting chords theorem, or just the chord theorem, is a statement that describes a relation of the four line segments created by two intersecting chords within a circle. It states that the products of the lengths of the line segments on each chord are equal. It is Proposition 35 of Book 3 of Euclid's Elements.
In Euclidean geometry, the intersecting secants theorem or just secant theorem describes the relation of line segments created by two intersecting secants and the associated circle.
The geometrical idea of the tangent line as the limit of secant lines serves as the motivation for analytical methods that are used to find tangent lines explicitly. The question of finding the tangent line to a graph, or the tangent line problem, was one of the central questions leading to the development of calculus in the 17th century.
Secant-, chord-theorem. For the intersecting secants theorem and chord theorem the power of a point plays the role of an invariant: . Intersecting secants theorem: For a point outside a circle and the intersection points , of a secant line with the following statement is true: | | | | = (), hence the product is independent of line .
In geometry, an intersection is a point, line, or curve common to two or more objects (such as lines, curves, planes, and surfaces). The simplest case in Euclidean geometry is the line–line intersection between two distinct lines , which either is one point (sometimes called a vertex ) or does not exist (if the lines are parallel ).
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