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        riddles >> medium >> Third Circle
        
(Message started by: Sir Col on Apr 14^th, 2004, 7:28am)

Title: Third Circle
Post by Sir Col on Apr 14^th, 2004, 7:28am

An arbitrary point, T, is chosen on line segment, AB.
Circles AT and BT are drawn, with radii r and R respectively.
Two distinct points, P and Q, are chosen, one on each circle such that PQ is tangential to both circles.
The midpoint of PQ is M.
Circle MP is drawn.
Find the radius of MP in terms of r and R.

What if, instead of having one common point on AB, S and T are located on AB so that circles AS and BT are drawn?

Using straight edge and compass, how would you locate P and Q?

Title: Re: Third Circle
Post by Noke Lieu on Apr 14^th, 2004, 5:08pm

First thought was is MP centred on m or p? Damn, I can be stupid sum times. :D
Hope its just a temporary thing :-/
[hide]
Radius of MP is pq/2

pq is determined by (R+r)[sup2]-(R-r)[sup2]=pq[sup2]
where R+r is AB

here's my thinking...(very similar to the rubber band question)
[/hide] ;)

Title: Re: Third Circle
Post by Noke Lieu on Apr 14^th, 2004, 5:14pm

Just before I get on with some work ::)
am wondering if its assumed that AS<AT, as in A is closer to S than T? I don't know if its relevant, but seemed like a cheeky point....

Title: Re: Third Circle
Post by Sir Col on Apr 14^th, 2004, 5:46pm

Great work, once more!

You're right, and it is an amazing coincidence, but this problem is similar to the "Wrapped Circles" problem.

I think you'll be quite impressed if you work out the radius of MP in terms of r and R; it has quite an amazing relationship with the two lengths. What is even more fascinating is the underlying discovery I made, and which I hope will emerge through further exploration. Which links very nicely with the answer to your last question... remarkably, it doesn't matter!