Minimum perimeter of quadrilateral
Given a cyclic quadrilateral ABCD where AB=20, BC=40, CD=30, and DA=50. Construct another quadrilateral P, Q, R, S where P is along AB, Q is along BC, R is along CD, and S is along DA that will produce the minimum perimeter.
http://www.geogebra.org/forum/viewtopic.php?f=2&t=34135
If O is the intersection of the diagonals AC and BD, und P, Q,R, S are the feet of the perpendiculars of O on the sides AB, BC, CD, DA, respectively, --> PQRS is the quadrilateral of minimum perimeter inscribed in ABCD - It can be proven.
Using Geogebra this problem is solved here by computing the extrema of functions of 4 variables. The problem has infinitely many solutions.