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?

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