Maximizing the Area of a Quadrilateral

 

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We saw in the previous section how to calculate the area of a general quadrilateral.   The basic formula is (2.1)

 

and it refers to the picture:

 

 

We also saw that once four lengths: a,b,c,d  are given in order, then  any quadrilateral that is constructed from those lengths satisfies the condition that the points (T,P) associated with them lie on a curve.  The equation (2.2) for that curve, once again is:

 

 

          What we would like to learn is (for a given sequence of lengths:  a,b,c, and d)  which are the quadrilaterals whose points (T,P) are such that they give a local maximum value for the area when T and P are  plugged into the area formula? 

 

          We will now show with a calculation, and then demonstrate in an experiment that this can only happen when   .  Once we know this, it will be a straightforward matter to see what the largest area of any quadrilateral associated to a,b,c, and d are. 

 

           If we look at equation (2.2) for a, b, c, and d held fixed,  we will see that the curve it defines is just the set of points in the (T,P) plane at which the function:

 

                       

 takes the constant value:   .

 

           Now the area function that we want to maximize is also a function  of T and P:  say

 

 

   We drop the absolute values because we are only going to consider T and P

   that are strictly between 0  and .

 

             Now, as long as we restrict T and P to be between 0 and  it is fairly clear that we may express P locally as a function of T.  That is, for any pair  (T,P)  between 0 and  that come from a quadrilateral, for  sufficiently close to T, there will correspond a unique  such that (T',P') comes from a nearby quadrilateral. 

 

Thus we may write (locally)   where P is a differentiable function of T in the vicinity of points on the curve (2.2).

 

            While this reasoning is not rigorous, it can be made so by using the Implicit Function Theorem of the calculus.  We do not have to resort to it here because it is in this case obvious, given the geometric basis for (2.2).  In fact, it is what made it possible for the computer to deform the quadrilaterals of the previous exercises. 

 

       Since F(T,P) is constant along the curve, this implies that

 

 

is constant as  a function of T, or that     = 0  for all T.   And this is easily seen to imply

(3.1)

 

     Continuing with this idea, we may write the area function G as a function of T

  alone (locally) and so we want to find local maxima for:

 

 

        Since all considerations are local here, we find that a necessary condition for (T, P(T)) to correspond to a quadrilateral that maximizes area locally is that   .  Finally, this means that, at such T,  

  

(3.2)

 

 

 

        From (3.1), it follows that the vector (in the T-P plane) 

 

is perpendicular to the (nonzero)  vector

 

And from (3.2) it follows that the vector

 

 

is also perpendicular to 

 

  Theorem:

     We have therefore the necessary condition, assuming that a,b,c,d are all nonzero:  (T,P) is a point at which area is maximized  (for 0 < T , P < pi)   only if the vectors:

     

            and     are parallel.

 

   Exercise:  Show that if the condition of the theorem is satisfied, then

 

                     1)  

                      and so

                     2) 

 

    Exercise:  Show that for each quadrilateral determining (T, P) as above, the extremum cannot be a local minimum.  Show also that for convex quadrilaterals, such a point must in fact be a local maximum.  Assume that everything is differentiable.  

 

         The argument of the preceding few paragraphs is made more nicely (and concisely) using the Method of Lagrange Multipliers from the Calculus.

 

         We are interested in the result because it says that those quadrilaterals constructed from fixed a,b,c, and d, whose area is locally maximized must satisfy the condition that for a pair of opposite angles T and P,   (That is, they are supplementary).  Now, of course, the other pair of opposite angles must also be supplementary  (Why?). 

    

         The following exercise explains where we are going with these observations.

 

       Exercise:  Show that a quadrilateral has opposite angles supplementary if and only if it is embeddable in a circle. 

      

            We will see in the next section that for such quadrilaterals, it is a straightforward matter to calculate their area.  And we shall see there the proper setting for Heron's formula.

 

            Before getting into those things, let's explore this idea that the quadrilaterals which, for fixed values of a,b,c and d, maximize the area are in fact embeddable in circles.  The laboratory following will give us the opportunity to do that.