help

Can you explain more clearly?

What does "O1O2" MEAN?

Does it mean distance?

Here's my attempt at this  tricky problem

Note that  angle ABC  is inscribed in the  first circumcircle

Let the circumcenter   =  M  = O1

Then  angle   AMC is a  central angle intercepting the same arc  as angle ABC  so its measure =  120°

And triangle AMC  is isoceles  with AM = CM

And angles  MAC and MCA =   (180 - 120) / 2  =    30°

Using some trig  we can find circumradius  AM  as

AM / sin 30°   =   12 / sin 120°

AM / (1/2)  =  12  / ( [sqrt 3 ] / 2 ) 

AM  =  12/sqrt 3  =   4sqrt 3

Now triangle   ACD  is obtuse  and its circumcenter will be exterior to side AD

Angle   ADC will be inscribed in  the circumcircle of this triangle

Let N be the circumcenter O2

Then angle  ANC  is a central angle intercepting the same  arc as angle ADC so its measure = 60°

And triangle ANC  is isosoceles with AN = CN

So  angles   CAN and ACN   = (180 - 60) / 2  =  60°

So triangle ANC  is equilateral  with circumradius   AN  =   12

Let  A =  (0,0)

Then  M =  O1  has the coordinates   (   4sqrt (3) cos 30° , 4sqrt (3) sin 30°)  =  (6 , 2sqrt (3) )

And  N =  O2  has the coordinates   ( 12 cos (-60°) , 12 sin (-60°) )  =   ( 6  , -6sqrt (3) )

Assuming that    O1 O2    is the distance between the  circumcenters we have  that

O1 O2  =   2sqrt (3)   -  (-6sqrt (3) )   =    8 sqrt (3)