Let O and ROandR be the center and the radius of the big circle and O' and r the center and the radius of the small circle, respectively, as shown in the figure.
=> Obviously, R=OE=OG=2
recall that tangent segments to a circle from an external point are equal in length, => EB=BF=1, CG=CF,
=> DeltaOBE and DeltaOBF are congruent, and DeltaOCF and DeltaOCG are congruent.
Let angleBOE=alpha, => angleBOF=alpha,
let angleCOF=beta, => angleCOG=beta
=> angleEOG=2(alpha+beta)=180^@,
=> alpha+beta=90^@
=> angleOCF=angleOCG=90-beta=alpha,
=> tanalpha=(BF)/(OF)=1/2,
=> sinalpha=1/sqrt5
Now consider DeltaOHO',
sinalpha=(OH)/(OO'),
=> 1/sqrt5=(2-r)/(2+r)
=> sqrt5(2-r)=2+r
=> 2sqrt5-2=r(1+sqrt5)
=> r=(2sqrt5-2)/(1+sqrt5)=3-sqrt5~~0.7639 units