# Circle geometry: nonparallel tangent and secant problem

If secant and the tangent of a circle intersect at a point outside the circle then prove that the area of the rectangle formed by the two line segments corresponding to the secant is equal to the area of the square formed by the line segment corresponding to the tangent
I find this question highly confusing. I do not know what this means. If you could please explain that to me and solve it if possible.

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Do you have any thoughts at all about the question?? –  Rustyn Dec 21 '12 at 18:00
I decided to change the title ("How do I solve this?") to something that distinguishes it from the other 50,000+ questions. Please always use an informative, descriptive title. That is, unless you are trying to camouflage your question from being seen. –  rschwieb Dec 21 '12 at 18:04
I don't understand: e have a secant and a tangent to a given circle: what "rectangle forme by the two segment corresponding to the secant" are we talking about here?? Is that perhaps the cord and the exterior part of the secant or what? –  DonAntonio Dec 21 '12 at 18:05
So far I'm the only one who's up-voted this question or the answers other than my own. –  Michael Hardy Dec 22 '12 at 2:59

The best reading I can find is suggested by DonAntonio. We are asked to prove $|AB|^2=|AC||AD|$

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Is there any reason that you chose AD over CD? I think Don and I were referring to CD. I suppose knowing the solution to the problem would clear this question up, of course... –  rschwieb Dec 21 '12 at 18:36
@rschwieb: because it is not true with CD. Imagine the secant rotating to become very close to the tangent. CD gets very small and AC gets close to AB. But you have some hope with AD-both AD and AC get close to AB. –  Ross Millikan Dec 21 '12 at 18:53
OK, that's the "knowing the solution" component I was referring to. This is really not possible to guess from the OP... –  rschwieb Dec 21 '12 at 19:32

Others have answered this, but here is a source of further information:

http://en.wikipedia.org/wiki/Power_of_a_point

Here's a problem in which the result is relied on:

http://en.wikipedia.org/wiki/Regiomontanus%27_angle_maximization_problem#Solution_by_elementary_geometry

The result goes all the way back (23 centuries) to Euclid (the first human who ever lived, with the exception of those who didn't write books on geometry that remain famous down to the present day):

http://aleph0.clarku.edu/~djoyce/java/elements/bookIII/propIII36.html

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Very cool. That is a very interesting distinction you give to Euclid, as well :) –  rschwieb Dec 21 '12 at 18:51