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# help

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Let be the circumcenter of triangle and let be the circumcenter of triangle Find  Apr 14, 2022

#1
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Can you explain more clearly?

What does "O1O2" MEAN?

Does it mean distance?

Apr 15, 2022
#2
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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)   Apr 15, 2022