# 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.