Tangent-Secant
A secant of a circle is a line connecting two points on the circle. A tangent to a circle is a line that intersects a circle exactly once. When a nonparallel tangent and secant are given, their intersection point satisfies several interesting properties.
These properties are very important when dealing with tangents in general, as they are almost always coupled with other secants, allowing these results to apply.
Viewing a Tangent as a Secant
Let be a circle, be a line tangent to at , be a secant of , and be the intersection of and . Though intersects at only one point, it can also be viewed as a secant between two infinitely close points. In other words, if is a line tangent to a circle at , can also be viewed as the secant .
This is a particularly important change of perspective, as it allows the same properties that applied to two secants to also apply to the case of a tangent and a secant. In particular, a key result is that
which follows immediately from the characterization above; it is equivalent to the equal angles formed by the diagonals of cyclic quadrilaterals.
Power of a Point
A secant of a circle is a line connecting two points on the circle. A tangent to a circle is a line that intersects a circle exactly once. When a nonparallel tangent and secant are given, their intersection point satisfies a key property: the third case of the power of a point theorem.
Let be a circle, be a line tangent to at , be a secant of , and be the intersection of and . Then
This is also a consequence of viewing a tangent as a secant between two infinitely close points, as in the previous section.