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.



  1. 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.

  2. Tanya Khovanova's Math Blog » Blog Archive » A Quadrilateral in a Rectangle Solution:

    […] recently posted A Quadrilateral in a Rectangle […]

  3. Carnival 170: A Plethora of Playful Math – Denise Gaskins' Let's Play Math:

    […] Khovanova poses a nice Quadrilateral in a Rectangle puzzle (solution […]

Leave a comment