# 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.

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