# Concave quadrilateral with integer sides and integer diagonals

Which is the concave quadrilateral with integer sides and integer diagonals with the smallest possible perimeter? This puzzle is my own creation.

I shamelessly exploited the fact that there is no tag and wrote a little proggy.

The best non-intersecting non-degenerate it finds has

sides 6,8,19,17 and diagonals 4 and 22. The perimeter is 50.

Picture

Left: solution as described, right: same set of vertices but reassigned so the longer diagonal is inside, here the perimeter sums to 51

• Are you sure? I found a smaller perimeter with my program. Jan 28, 2021 at 19:48
• @Vepir no, I'm never sure with my programs ;-) Jan 28, 2021 at 19:50
• Never mind, my bad. It was a "near example" due to mishandling float precision. After patching it, I can confirm that what you found is smallest. Jan 28, 2021 at 20:14
• Nice! I found the same solution by computer search so I am quite confident that it is indeed the solution. The short diagonal is inside the quadrilateral, the long outside. What if we request to have the long diagonal inside? My best solution has perimeter 51 in this case. Jan 28, 2021 at 20:14
• @PaulPanzerYou got it almost. Just connect the 4 corners in a different way... Jan 28, 2021 at 21:22