SECTION 1.3 Circle Geometry 33
Corollary. (Steiner’s Theo-
rem)We are given the a circle, and
secant lines(PA)and(PC), where
(PA)also intersects the circle atB
and where(PC)also intersects the
circle atD.
PA×PB = PC×PD.Proof. Note that only the case
in whichP is interior to the circle
needs proof. However, since angles
CBP̂ andPDÂ open the same are,
they are equal. Therefore, it follows
instantly that 4 PDA ∼ 4PBC,
from which the result follows.
The product PA×PB of the distances from the point P to the
points of intersection of the line through P with the given circle is
independent of the line; it is called the power of the point with
respect to the circle. It is customary to usesigned magnitudeshere,
so that the power of the point with respect to the circle will benegative
precisely whenP is insidethe circle. Note also that the power of the
pointP relative to a given circleC is a function only of the distance
fromP to the center ofC. (Can you see why?)
The second case of Steiner’s theorem is sometimes called the “Inter-
secting Chords Theorem.”
Exercises
- In the complex plane, graph the equation|z+ 16|= 4|z+ 1|. How
does this problem relate with any of the above?