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Next to the intersecting chords theorem and the tangent-secant theorem, the intersecting secants theorem represents one of the three basic cases of a more general theorem about two intersecting lines and a circle - the power of point theorem.
For example, if K is a set of 50 points arranged on a circle in the Euclidean plane, a line joining two of them would be a 2-secant (or bisecant) and a line passing through only one of them would be a 1-secant (or unisecant). A unisecant in this example need not be a tangent line to the circle.
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.
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 .
Thus, for 3-dimensional spaces, one needs to show that (1*) every point lies in 3 distinct planes, (2*) every two planes intersect in a unique line and a dual version of (3*) to the effect: if the intersection of plane P and Q is coplanar with the intersection of plane R and S, then so are the respective intersections of planes P and R, Q and S ...
a) different tangent lines (transversal intersection, after transversality), or b) the tangent line in common and they are crossing each other ( touching intersection , after tangency ). If both the curves have a point S and the tangent line there in common but do not cross each other, they are just touching at point S .
If a chord were to be extended infinitely on both directions into a line, the object is a secant line. The perpendicular line passing through the chord's midpoint is called sagitta (Latin for "arrow"). More generally, a chord is a line segment joining two points on any curve, for instance, on an ellipse.
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 ...