What is the Value of Angle MBC in the Triangle-Square Configuration?

[anemone]

Here is this week's POTW:

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Let $ABC$ be a right triangle with right angle at $B$. Let $ACDE$ be a square drawn exterior to triangle $ABC$. If $M$ is the center of this square, evaluate $\angle MBC$.-----

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[anemone]

Congratulations to the following members for their correct solution!(Cool)

1. castor28
2. kaliprasad

Solution from castor28:

Spoiler

View attachment 8761
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Let $P$ be the midpoint of $AC$. As $PM=AP$, $M$ lies on a circle with diameter $AC$.
On the other hand, as $\angle ABC=90\mbox{°}$, the point $B$ also lies on that circle.
As the arc $CM$ equals $90\mbox{°}$, the inscribed angle $\angle MBC$ equals $45\mbox{°}$.