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Ana and Bob are playing the following game.
[*] First, Bob draws triangle

ABC

and a point

P

inside it. [*] Then Ana and Bob alternate, starting with Ana, choosing three different permutations

\sigma_1

\sigma_2

and

\sigma_3

\{A, B, C\}

. [*] Finally, Ana draw a triangle

V_1V_2V_3

.
For

i=1,2,3

, let

\psi_i

be the similarity transformation which takes

\sigma_i(A), \sigma_i(B)

and

\sigma_i(C)

V_i, V_{i+1}

and

X_i

respectively (here

V_4=V_1

) where triangle

\Delta V_iV_{i+1}X_i

lies on the outside of triangle

V_1V_2V_3

. Finally, let

Q_i=\psi_i(P)

. Ana wins if triangles

Q_1Q_2Q_3

and

ABC

are similar (in some order of vertices) and Bob wins otherwise. Determine who has the winning strategy.

Problem Statement