32 CHAPTER 1 Advanced Euclidean Geometry
1.3.2 Steiner’s theorem and the power of a point
Secant-Tangent Theorem. We
are given the a circle, a tangent line
(PC) and a secant line(PA), where
Cis the point of tangency and where
[AB]is a chord of the circle on the
secent (see the figure to the right.
Then
PC^2 = PA×PB.Proof. This is almost trivial;
simply note that PCÂ and ABĈ
open the same angle. Therefore,
4 PCA∼ 4PBC, from which the
conclusion follows.
There is also an almost purely algebraic proof of this result.^7The following is immediate.
(^7) If the radius of the circle isrand if the distance fromPto the center of the circle isk, then
denotingdthe distance along the line segment to the two points of intersection with the circle and
using the Law of Cosines, we have thatr^2 =k^2 +d^2 − 2 kdcosθand sodsatisfies the quadratic
equation
d^2 − 2 kdcosθ+k^2 −r^2 = 0.
The product of the two roots of this equation isk^2 −d^2 , which is independent of the indicated angle
θ.
P
d
r
k
θ