problem on circle theorems

• Mar 14th 2009, 02:11 AM
smmmc
problem on circle theorems
PQ is a variable chord of the smaller of two fixed concentric circles.
PQ produced meets the circumference of the larger circle at R. Prove that the product
RP.RQ is constant for all positions and lengths of PQ.

How would you prove this?

thanks
• Mar 14th 2009, 02:41 AM
running-gag
Hi

First you can take some simple examples to have an idea of the value of the constant
The easiest one is to take PQ as a diameter of the smaller circle

Then considering the generic case
http://nsa06.casimages.com/img/2009/...2635770951.jpg

Let O be the center of the two circles and I the center of [PQ]

$\displaystyle \overrightarrow{RP}\cdot\overrightarrow{RQ}=RP\:RQ$

$\displaystyle \overrightarrow{RP}\cdot\overrightarrow{RQ}=\left( \overrightarrow{RI}+\overrightarrow{IP}\right)\cdo t\left(\overrightarrow{RI}+\overrightarrow{IQ}\rig ht)$

Expanding and simplifying

$\displaystyle \overrightarrow{RP}\cdot\overrightarrow{RQ}=RI^2-IP^2$

$\displaystyle \overrightarrow{RP}\cdot\overrightarrow{RQ}=(RO^2-OI^2)-IP^2=RO^2-(OI^2+IP^2)=RO^2-OP^2$

(Properties of right angle triangles)

Therefore RP.RQ is equal to the difference of the squares of the 2 radii