# When Does This Point Not Exist?

Geometry Level 5

Let $$\triangle ABC$$ be an acute angles triangle with $$\angle BAC = 80^{\circ}.$$ Let $$M$$ be the midpoint of $$BC,$$ and let $$O$$ be the circumcenter of $$\triangle ABC.$$ Suppose there exists no point $$X$$ in the plane of $$\triangle ABC$$ which satisfies the following conditions.

• $$X \neq O, X \neq A$$
• $$\angle BAX = \angle CAM$$
• $$\angle AXO = 90^{\circ}$$

Let $$\angle ABC = N^{\circ}.$$ Find $$\dfrac{N}{2}.$$

Details and assumptions
- The diagram shows the point $$X,$$ which shouldn't exist.
- The first condition means $$X$$ must be different from $$O$$ and $$A.$$

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