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Circle's Limit (Posted on 2006-01-13) Difficulty: 2 of 5
Let C1 be a circle with center A and radius 1. Let m be a line tangent to circle C1 at point B. Let C2 be a circle with center B intersecting line m at points C and D and circle C1 at points E and F (points labeled such that C and E are on the same side of line AB). Let line CE intersect line AB at point G. As the radius of circle C2 shrinks to zero, does the length of BG approach a limit? If yes, then what is its value?

See The Solution Submitted by Bractals    
Rating: 3.0000 (3 votes)

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Solution No angles Comment 4 of 4 |

Setting B = (0,0) and A = (1,0), then C = (0, r) where r is the radius of C2.

Now E is on the intersection of the circles C1: (x-1)^2 + y^2 =1 and C2: x^2 + y^2 =r^2.

Now with a bit of algebra we calculate that the line through CE meets the x-axis at x = 2 + Sqrt(4 - r^2) for r>0.

And trivially gives the solution BG = x -> 4 as r->0.


  Posted by goFish on 2006-01-14 18:34:08
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