A Quadrilateral in a Rectangle
A new geometry problem from my friend, Alexander Karabegov.
Puzzle. A convex quadrilateral is inscribed in a rectangle with exactly one quadrilateral’s vertex on each side of the rectangle. Prove that the area of the rectangle is twice the area of the quadrilateral if and only if a diagonal of the quadrilateral is parallel to two parallel sides of the rectangle.
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Lazar Ilic:
If a diagonal is parallel then it follows by drawing the triangles and bh/2 area splitting each subrectangle in half. The other direction follows from a relatively simple smoothing argument. If BD is not parallel then the area function is monotone and nonconstant linear as we shift point A along its edge and in particular obtains its unique 1/2 valuation when AC is parallel. More formally this can be observed in the literal definitional derivative d/da of the area (a*(1-b)+b*(1-c)+c*(1-d)+d*(1-a))/2 in terms of their locations along their respective edges.
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