MathDB

Let

A

and

B

points in the plane and

C

a point in the perpendiclar bisector of

AB

. It is constructed a sequence of points

C_1,C_2,\dots, C_n,\dots

in the following way:

C_1=C

and for

n\geq1

, if

C_n

does not belongs to

AB

, then

C_{n+1}

is the circumcentre of the triangle

\triangle{ABC_n}

.
Find all the points

C

such that the sequence

C_1,C_2,\dots

is defined for all

n

and turns eventually periodic.
Note: A sequence

C_1,C_2, \dots

is called eventually periodic if there exist positive integers

k

and

p

such that

C_{n+p}=c_n

for all

n\geq{k}

Problem Statement