Question

In a quadrilateral ABCD angle A+angle D=90degree prove that ACsquare +BDsquare =ADsquare +BCsquare.

Answer 1

In triangle AEC ,

AE^2 + EC^2 = AC^2 --------------- [1]

In triangle BED ,

BE^2 + ED^2 =BD^2 --------------- [2]

Adding [1] and [2] ,

AC^2 + BD^2 = AE^2 + EC^2 + BE^2 + ED^2 --------------------- [3]

In triangle AED ,

AE^2 + ED^2 = AD^2 ---------------- [4]

In triangle BEC ,

BE^2 + EC^2 = BC^2 ---------------- [5]

Adding [4] and [5] ,

AD^2 + BC^2 = AE^2 + ED^2 +BE^2 + EC^2 = AE^2 + EC^2 + BE^2 + ED^2 ------------------------- [6]

From [3] and [6] ,

AC^2 +BD^2 = AD^2 + BC^2

Hence proved.