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.
Share:
![Share on Facebook Facebook](https://blog.tanyakhovanova.com/wp-content/plugins/social-media-feather/synved-social/image/social/regular/48x48/facebook.png)
![Share on Twitter twitter](https://blog.tanyakhovanova.com/wp-content/plugins/social-media-feather/synved-social/image/social/regular/48x48/twitter.png)
![Share on Reddit reddit](https://blog.tanyakhovanova.com/wp-content/plugins/social-media-feather/synved-social/image/social/regular/48x48/reddit.png)
![Pin it with Pinterest pinterest](https://blog.tanyakhovanova.com/wp-content/plugins/social-media-feather/synved-social/image/social/regular/48x48/pinterest.png)
![Share on Linkedin linkedin](https://blog.tanyakhovanova.com/wp-content/plugins/social-media-feather/synved-social/image/social/regular/48x48/linkedin.png)
![Share by email mail](https://blog.tanyakhovanova.com/wp-content/plugins/social-media-feather/synved-social/image/social/regular/48x48/mail.png)
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.
29 November 2023, 4:54 amTanya Khovanova's Math Blog » Blog Archive » A Quadrilateral in a Rectangle Solution:
[…] recently posted A Quadrilateral in a Rectangle […]
2 December 2023, 4:43 pmCarnival 170: A Plethora of Playful Math – Denise Gaskins' Let's Play Math:
[…] Khovanova poses a nice Quadrilateral in a Rectangle puzzle (solution […]
5 December 2023, 9:28 am