Looking for an inscribed quadrangle in a rectangle with mini

[Guest]

I thought up an other math problem, and hope you all will find it interesting.

I'm supposed to find a quadrangle of the smallest perimeter possible inscribed in a rectangle. The inscribed quadrangle has each of its four vertices on another side of the rectangle.

Let's call the rectangle ABCD and let the shorter side be a and the larger b. Let's call the quadrangle PQRS.

So for example P lies on AB, Q on BC, R on CD, S on AD.

Have you an idea how can I solve this problem. Please help me.

[HallsofIvy]

You can, without loss of generality, take one vertex of the rectangle as the origin of a coordinate system with positive x and y axes along two sides of the rectangle. So the equations of the four lines are x= 0, y= 0, x= L, and y= W where L and W are the length and width of the rectangle. So the vertices can be written as (0, a), (b, 0), (c,W), and (L, d). The perimeter is sqrt{(b- c)^2+ W^2}+ sqrt{(c- L)^2+ (W- d)^2}+ sqrt{(L- b)^2+ d^2}+ sqrt{a^2+ b^2}. Take the partial derivatives of that with respect to a, b, c, and d, set the four derivatives equal to 0, and solve those four equations for a, b, c, and d (W and L are constants).