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?

Geometer's sketchpad implies the solution is 4.
Here's a possible analytic proof (but I feel like I broke the rules somewhere.  Or is this the calculus?)

Let c1 be (x-1)^2 + y^2 =1
Let c2 be x^2 + y^2 = r^2

C = (0,r)
E = (-r^2/2, r/2*sqrt(4-r^2))

[algebra omitted]
The line CE hits the x axis at x = -r^2/(2-sqrt(4-r^2))

2x - x*sqrt(4-r^2) = -r^2
r^2 + 2x = x*sqrt(4-r^2)
r^4 + 4xr^2 + 4x^2 = 4x^2 - x^2*r^2
x^2 + 4x + r^2 = 0 [it is legal to divide by r^2?]

If r=0 then

x^2 + 4x =0
Which has solutions

x=-4 or x=0

The 0 is extraneous, the x=-4 implies a distance of 4 from the origin.

Posted by Jer on 2006-01-13 10:29:30