A new Way the Area of Trapezium obtain by PiyushGoel Lot of mathematicians have proved Pythagoras theorem in their own ways. If you google it you will indeed found hundred of ways.
Meanwhile I was also sure that maybe one day I could find something new out of this incredible Pythagoras theorem and Recently I got something which I would like to share with you.
To Prove: Deriving the equation of area of trapezium using Arcs
Proof: There is a triangle ABC with sides a b and c as shown in the figure.
Area of ? BCEG = Area of ? BDC +Area of ¦ DCEF + Area of ? EFG
c^2=ac/2+ Area of ¦ DCEF + (c-b) c/2
(2c^2– ac –c^2+ bc )/2=Area of ¦ DCEF
(c^2– ac+ bc )/2=Area of ¦ DCEF
c(c– a+ b)/2=Area of ¦ DCEF
Area of ¦ DCEF=BC(DE+CF)/2