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 \(O\) be a circle, \(\ell\) be a line tangent to \(O\) at \(T\), \(AB\) be a secant of \(O\), and \(P\) be the intersection of \(AB\) and \(\ell\). Though \(\ell\) intersects \(O\) at only one point, it can also be viewed as a secant between two infinitely close points. In other words, if \(\ell\) is a line tangent to a circle \(O\) at \(T\), \(\ell\) can also be viewed as the secant \(TT\).
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
\[\angle PTB = \angle TAP\]
which follows immediately from the characterization above; it is equivalent to the equal angles formed by the diagonals of cyclic quadrilaterals.
\(P\) is a point outside of circle \(\Gamma\). The tangent from \(P\) to \(\Gamma\) touches at \(A\). A line from \(P\) intersects \(\Gamma\) at \(B\) and \(C\) such that \( \angle ACP = 120^\circ \). If \(AC = 16\) and \(AP = 19\), then the radius of \(\Gamma\) can be expressed as \( \frac {a \sqrt{b} }{c} \), where \(b\) is an integer not divisible by the square of any prime and \(a, c\) are coprime positive integers. What is \( a + b + c\)?
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 \(O\) be a circle, \(\ell\) be a line tangent to \(O\) at \(T\), \(AB\) be a secant of \(O\), and \(P\) be the intersection of \(AB\) and \(\ell\). Then
\[ PT^2 = PA \cdot PB\]
This is also a consequence of viewing a tangent as a secant between two infinitely close points, as in the previous section.
