Maximizing area of a Quadrilateral with given side lengths

We seek a quadrilateral with side lengths a,b,c,d (specified below) that maximizes area.

a=5
b=E
c=Pi
d=3

If s and t are the angles formed by the sides of length a and d and of length b and c, respectively, then the area of the quadrilateral is a function A[s, t] subject to the constraint ρ[s, t] = 0, where A and ρ are as follows.

A[s_,t_]=(a d Sin[s]+b c Sin[t])/2
ρ[s_,t_]=2(a d Cos[s]-b c Cos[t])-a^2 - d^2+b^2 + c^2

We showed in class that the maximum possible area is

maxA = N[Sqrt[(a d + b c)^2-(a^2+d^2-b^2-c^2)^2/4]/2]

We see geometrically what’s happening when A is maximized in the following picture which shows 50 different level sets of A together with the curve ρ=0.  A has a constrained maximum at the point where its level set is tangent to that of the constraint curve.

plot1=ContourPlot[{ρ[s,t]==0},{s,0,Pi},{t,0,Pi}];
plot2=ContourPlot[A[s,t],{s,0,Pi},{t,0,Pi},Contours->50];
Show[plot2,plot1]

Mathematica challenge: get Mathematica to draw the area maximizing quadrilateral together with the circumscribing circle.