Points $A, B, C, M$ are such that $B, M, C$ lie on a line, $A B=A C=353$, and $B M=$ $M C$. Point $I$ is on segment $A M$ such that the circle centered at $I$ through $M$ has radius 106 and is tangent to $A B$ and $A C$. Given that there are two possible areas of right triangle $A M B$, find the larger one.
The two configurations that work are the configuration when $B=C$ and angle $A$ is right and the configuration when $B=C$ and angle $B$ is right. In the second case, the height from $M$ is $2 \cdot 106$, which is larger than in the other case, The answer is thus $106 \cdot 353=37418$.
![](https://cdn.mathpix.com/cropped/2024_04_17_d3ed8ef337b2bc5b6a3fg-1.jpg?height=362&width=1306&top_left_y=1635&top_left_x=474)