1976 IMO Problems/Problem 1
In a convex quadrilateral (in the plane) with the area of the sum of two opposite sides and a diagonal is . Determine all the possible values that the other diagonal can have.
Label the vertices , , , and in such a way that , and is a diagonal.
The area of the quadrilateral can be expressed as , where and are altitudes from points and onto . Clearly, and . Hence the area is at most .
The quadratic function has its maximum for , and its value is .
The area of our quadrilateral is . This means that we must have . Also, equality must hold in both and . Hence both and must be perpendicular to . And in any such case it is clear from the Pythagorean theorem that .
Therefore the other diagonal has only one possible length: .
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