Question

用勾股法证明平行四边形对角线的平方和等于各边平方和

Answer 1

平行四边形ABCD,过A、B作CD的垂线分别交于E、F

AC^2=AE^2+(CD-DE)^2

BD^2=BF^2+(CD+CF)^2

AE=BF

CF=DE

AC^2+BD^2=AE^2+(CD-DE)^2+BF^2+(CD+CF)^2=2AE^2+(CD-DE)^2+(CD+DE)^2

=2AE^2+2CD^2+2DE^2=2AD^2+2CD^2