Editing Secant line (section)

==Circles==
{{more|Circle#Chord}}
[[Image:CIRCLE LINES-en.svg|thumb|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 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 [[Euclid's Elements|his treatment]], are usually proved.

For example, ''Theorem (Elementary Circular Continuity)'':<ref>{{citation|first=Gerard A.|last=Venema|title=Foundations of Geometry|year=2006|publisher=Pearson/Prentice-Hall|page=229|isbn=978-0-13-143700-5}}</ref> If <math>\mathcal{C}</math> is a circle and <math>\ell</math> a line that contains a point {{mvar|A}} that is inside <math>\mathcal{C}</math> and a point {{mvar|B}} that is outside of <math>\mathcal{C}</math> then <math>\ell</math> is a secant line for <math>\mathcal{C}</math>.

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:<ref>{{citation|first=Harold R.|last=Jacobs|title=Geometry|year=1974|publisher=W. H. Freeman & Co.|page=482|isbn=0-7167-0456-0}}</ref>
:If two secant lines contain chords {{math|{{overline|''AB''}}}} and {{math|{{overline|''CD''}}}} in a circle and intersect at a point {{mvar|P}} that is not on the circle, then the line segment lengths satisfy {{math|1=''AP''⋅''PB'' = ''CP''⋅''PD''}}.
If the point {{mvar|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.<ref>{{citation|first=Thomas L.|last=Heath|author-link=Thomas Little Heath|title=The thirteen books of Euclid's Elements (Vol. 2)|year = 1956|publisher=Dover|page=73}}</ref>