AMC2006高級組第30題

Theorem: The radii of four tangent circles in the plane satisfies the relation 2 ( 1/a^2 + 1/b^2 + 1/c^2 + 1/d^2 ) = (1/a+1/b+1/c+1/d)^2.

Proof: http://mathworld.wolfram.com/SoddyCircles.html

Plug in, and you're done.

設三圓圓心依次為(0,-1)，(0,2)，(x,y)，與三圓相切的圓半徑為k，圓心(a,b)，利用連心線長等於半徑和
則x^2+(y-2)^2=(2+3)^2，x^2+(y+1)^2=(1+3)^2
求得(x,y)=(4,-1),(-4,-1)
取(x,y)=(4,-1),同樣利用連心線長等於半徑和
a^2+(b-2)^2=(k+2)^2.......(1)
a^2+(b+1)^2=(k+1)^2.......(2)
(a-4)^2+(b+1)^2=(k+3)^2.....(3)
(1)-(2)得b=-k/3,(2)-(3)得a=-k/2+1
代入(1)得(23k-6)(k+6)=0，故k=6/23或k=-6
k=6/23，與三圓均外切的小圓半徑為6/23
k=-6仍有其幾何意義，代表與三圓均內切的大圓半徑為6