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Concept
Secant line
Summary
In geometry, a secant is a line that intersects a curve at a minimum of two distinct points.
The word secant comes from the Latin word secare, meaning to cut. In the case of a circle, a secant intersects the circle at exactly two points. A chord is the line segment determined by the two points, that is, the interval on the secant whose ends are the two points.
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 therefore contained in a unique secant line and each secant line determines a unique chord.
In rigorous modern treatments of plane geometry, results that seem obvious and were assumed (without statement) by Euclid in his treatment, are usually proved.
For example, Theorem (Elementary Circular Continuity): If is a circle and a line that contains a point A that is inside and a point B that is outside of then is a secant line for .
In some situations phrasing results in terms of secant lines instead of chords can help to unify statements. As an example of this consider the result:
If two secant lines contain chords and in a circle and intersect at a point P that is not on the circle, then the line segment lengths satisfy AP⋅PB = CP⋅PD.
If the point P lies inside the circle this is Euclid III.35, but if the point is outside the circle the result is not contained in the Elements. However, Robert Simson following Christopher Clavius demonstrated this result, sometimes called the intersecting secants theorem, in their commentaries on Euclid.
For curves more complicated than simple circles, the possibility that a line that intersects a curve in more than two distinct points arises. Some authors define a secant line to a curve as a line that intersects the curve in two distinct points. This definition leaves open the possibility that the line may have other points of intersection with the curve.
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