11.2. In a quadrilateral, the diagonals are perpendicular. A circle can be inscribed in it and a circle can be circumscribed around it. Can we assert that it is a square?
Problem 964
Official solution
Answer: No.
Solution. Consider in the circle the diameter and a chord perpendicular to it, not passing through the center (see figure).
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We will show that the quadrilateral satisfies the condition of the problem. For this, it is sufficient to prove that a circle can be inscribed in it. In a circle, the diameter bisects a chord perpendicular to it, so in triangle , the height is also a median, and this triangle is isosceles: . Similarly, . Since the sums of the opposite sides of the quadrilateral are equal, a circle can be inscribed in it.
## Comment.
7 points - complete solution.
3 points - correct example of the figure, but one of the properties (inscribed, circumscribed, or perpendicular diagonals) is not proven.
1 point - correct "picture".