Doesn't seem like Geometry!

Geometry Level 5

Consider \( 3 \) circles \( \omega_{1}, \omega_{2} \) and \( \omega_{3} \) on the Euclidean plane. Given that there exists a point \( M \) such that \( \omega_{1} \cap \omega_{2} \cap \omega_{3} = \{M\} \), and also \( \omega_{1} \cap \omega_{2} = \{A,M\} \), \( \omega_{1} \cap \omega_{3}=\{B,M\} \) and \( \omega_{2} \cap \omega_{3}=\{C,M\} \), consider the following statements :

1) There exist points \( D, E \) and \( F \) on \( \omega_{1}, \omega_{2} \) and \( \omega_{3} \) respectively such that the points \( (D,A,E), (E,C,F) \) and \( (D,B,F) \) are collinear and points \( (D,A,E,C,F,B,D) \) joined in the given order form a triangle.

2) There exist points \( X, Y \) and \( Z \) on \( \omega_{1}, \omega_{2} \) and \( \omega_{3} \) respectively such that \( \angle XBM = \angle YAM = \angle ZCM \).

3) Points \( (D,E,F) \) satisfy the properties of points \( (X,Y,Z) \) mentioned in 2).

4) \( M \) can lie only in the interior of \( \triangle DEF \).

5) \( M \) can lie only in the interior of \( \triangle ABC \).

6) \( \angle AMB = \angle AMC \).

Which of the above statements do you think always hold true ?

Enter the sum of the serial numbers of such statements as your answer.

NOTE :

  • Suppose you think that statements \( 4), 5) \) and \( 6) \) always hold true, you must enter the answer as \( 15 \).

  • Tuples are ordered lists, so be careful.

This is an original problem.
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