[ Relationship between the angle magnitude, arc length, and chord length ]

Inside the square $A B C D$, a point $M$ is chosen such that $\angle M A C=\angle M C D=\alpha$. Find the magnitude of the angle $A B M$.

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If point $M$ lies inside triangle $A B C$, then $\angle M A C < 45^{\circ} < \angle M C D$. It is also easy to verify that point $M$ cannot lie on the sides of triangles $A B C$ and $A C D$, so it must lie inside triangle $A C D$. In this case, $\angle A M C = 180^{\circ} - \angle M A C - (45^{\circ} - \angle M C D) = 135^{\circ}$. This means that point $M$ lies on the arc of a circle with radius $A B$ and center $B$. Therefore, by the inscribed angle theorem, $\angle A B M = 2 \angle A C M = 90^{\circ} - 2 \alpha$.

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