Let A, B and C represent the 3 boys standing equidistant from one another on a circle's boundary.
ABC form an equilateral triangle and the center of the circle O represents the circumcenter.
Draw a perpendicular OM from point O to side AB of the triangle.
Now, since a perpendicular from the center to a chord of a circle (i.e. AB here) bisects it.
So, MB = MA = 1/2 AB
Now, in triangle MOB, angle OMB = 90 and hence, it is a right angled triangle.
Furthermore,in triangle AOB and triangle COB
OB = BO (common)
AB = BC (sides of equilateral triangle)
AO = OC (radius)
triangle AOB is congruent to triangle COB (By SSS)
hence, angle(OBA) = angle(OBC) (By CPCT)
Now, all angles of an equilateral triangle are equal to 60 degree.
So, angle(OBA)+ angle(OBC) = 60
And since, they are equal, we get angle(OBA) = 30 degree.
Now in triangle MOB
MB = cos30 * OB
MB = sqrt(3)/2 * 20 = 10*sqrt(3)
Since, AB = 2*MB
So, AB = 20sqrt(3) cm
So,the length of the phone string = 20sqrt(3) cm