The large quadrilateral is a trapezoid with lower angles rig

The large quadrilateral is a trapezoid with lower angles right angles

Solution

Given fig: Let fig 1)

            

now divid this as

                    

   so triangle,1) area is:1/2ab

        triangle,2) area is:1/2 ab

        triangle,3)area is : 1/2c.c=1/2 c²

so total area of fig 1):1/2ab+1/2ab+c

                       A1= ab+c

and given: fig2)

                                 

the area fig 2) equal to 2 * area of fig 1)

                    so, 2(ab+1/2 c²)

                        A2=2ab+c²

    and: fig 3) is:

                    

                                         

      After removeing areas of four triangles from area of fig 2) we get area of fig 3)

       area of fig 3)

                          A3=2ab+c²- 2ab=c²

fig 4):

  

After removeing the area of squre of length (b-a) and dividing remaining area by 4 we get area of fig 4)

         area fig 4):

                   A4= [c²- (b-a)² ] /4                     ( area of trangle is:1/2 ab)

                         1/2 ab =[c²- (b-a)² ] /4

                   2ab=[c²- (b-a)² ]

                  2ab=c²- (b-a)²

                          2ab=c²-b²-a²+2ab

                  c2= a²+b²