Dual Isosceles trapezoidals
Isosceles trapezoidals T, T* are dual to each other if they have the same value of perimeter, area and diagonal length. They were found as counter examples to the false claim "Area, perimeter and diagonal length determine a convex quadrilateral uniquely". One interesting thing is that there are integer-lentgh such dual pair of trapozodals! And how to construct them is quite a challenge for now.