Problem of the Week

Take a square piece of paper with sides of length 1 and draw on it a quarter-circle of radius 1 with center at one of the vertices.  Fold the paper over so that the vertex which is the center of the circle just touches the quarter-circle.  See the figure on the left.  What is the largest and smallest areas that the shaded triangular region can have?

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ã2002 Alberto L. Delgado