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

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In the discussion, participants evaluate the angle MBC in a configuration involving a right triangle ABC and an exterior square ACDE. The problem highlights the geometric relationships between the triangle and the square, specifically focusing on the center M of the square. Members castor28 and kaliprasad provided correct solutions, demonstrating the mathematical reasoning involved. The evaluation of angle MBC reveals interesting properties of the triangle-square configuration. The thread emphasizes the importance of geometric understanding in solving such problems.
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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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Congratulations to the following members for their correct solution!(Cool)

1. castor28
2. kaliprasad

Solution from castor28:
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 $45\mbox{°}$.
 
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