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A B C D is a square E ,\ F ,\ G\ a n d\ ...

`A B C D` is a square `E ,\ F ,\ G\ a n d\ H` are points on `A B ,\ B C ,\ C D\ a n d\ D A` respectively, such that `A E=B F=C G=D Hdot` Prove that `E F G H` is square.

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`AE=BF=CG=DH`
Therefore, `BE=CF=DG=AH`
In `/_\AEH and /_\ BFE`
`AE=BF`(Given)
`/_A=/_B`(each equal to 90^0)
`AH=BE`(By SAS criterion of congruency)
`/_\AEH ~=/_\ BFE`
.`=>EH=BF`(By CPCT)
Similarity,`EH=HG=GF=FE`
Now,` /_AEH=/_BFE \and\ /_AHE=/_BEF`
But, `/_AEH+/_AHE=90^0`
`/_BFE+/_BFE=90^0`
`/_AEH+/_AHE+/_BFE+/_BFE=90^0+90^0`
`=>/_AEH+/_BEF+/_AEH+/_BEF=180^0`
`=>2(/_AEH+/_BEF)=180^0`
`=>/_AEH+/_BEF=90^0`
`/_HEF=90^0`
Similarly, /_EFG=/_FGH=/_GHE=90^0`
Therefore, EFGH is a square
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