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In the plane, three distinct non-collinear points

A,B,C

are marked. In each step, Ege can do one of the following:
[*] For marked points

X,Y

, mark the reflection of

X

across

Y

.
[*]For distinct marked points

X,Y,Z,T

which do not form a parallelogram, mark the center of spiral similarity which takes segment

XY

ZT

.
[*] For distinct marked points

X,Y,Z,T

, mark the intersection of lines

XY

and

ZT

.
No matter how the points

A,B,C

are marked in the beginning, can Ege always mark, after finitely many moves,
a) The circumcenter of

\triangle ABC

.
b) The incenter of

\triangle ABC

.
Proposed by Deniz Can Karaçelebi

Problem Statement