From Gogeometry

Using :

Additional points

Tangents to a circle

 •(1) AF=ra=r+EB+BF=r+a2
 •(2) AG=ra=r+DC+CG=r+a1
 •(1-2) a1=a2
 •(1+2) ra=r+a1=r+a2 => a1=a2=ra-r
 •Define X as the intersection of BC and circle ra 
 •Define Y as the intersection of BC and circle r
 •CD and CY tangents to circle r => CD=CY
 •BE and BY tangents to circle r => BE=BY
 •a=CY+BY=> (3) a=CD+BE
 •CG and CX tangents to circle ra => CG=CX
 •BF and BX tangents to circle ra => BF=BX
 •a=BX+CX=> (4) a=BF+CG
 •(3+4) 2a= CD+BE+ BF+CG=CD+CG+BE+ BF=a1+a2
 •=> a=a1=a2
 •Therefore a=a1=a2=ra-r