Gogeometry Problem 1372

Using :

Define a=AB=BC=AC

DC and DB tangents to Circle O1 => DG=DE=b
AD and AB tangents to Circle O1 => AG=AJ=d
DB and BA tangents to Circle O1 => BE=BJ=c

DB and BC tangents to Circle O2 => BF=BK=e
DC and BC tangents to Circle O2 => CI=CK=f

a=d+c=e+f
DF=b+c+e, DI=b+d+a+f
DC and DB tangents to Circle O2 => DF=DI
=> b+c+e=b+d+a+f => (a-d)+(a-f)=d+a+f
Therefore a=2d+2f

∠BAC=Π/3, ∠O1AG=∠O1AJ => ∠O1AG=∠O1AJ =Π/3
=>ΔAGO1 is similar to ΔAHB
In the same way ΔCIO2 is similar to ΔAHB

h=NB, a=AB=BC=AC => a=2h/sqrt(3)
In the same way, AG=d=r1/sqrt(3) and CI=f=r2/sqrt(3)

a=2d+2f => 2h/sqrt(3) =2 r1/sqrt(3) + 2 r2/sqrt(3)
Therefore h=r1+r2

Enigma