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Sir Col
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Third Circle
« on: Apr 14th, 2004, 7:28am » |
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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?
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Noke Lieu
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Re: Third Circle
« Reply #1 on: Apr 14th, 2004, 5:08pm » |
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First thought was is MP centred on m or p? Damn, I can be stupid sum times. 
Hope its just a temporary thing 
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)
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| « Last Edit: Apr 14th, 2004, 5:09pm by Noke Lieu » |
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Noke Lieu
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Re: Third Circle
« Reply #2 on: Apr 14th, 2004, 5:14pm » |
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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....
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Sir Col
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impudens simia et macrologus profundus fabulae
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Re: Third Circle
« Reply #3 on: Apr 14th, 2004, 5:46pm » |
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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!
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