Published on 29th March 2018, City: Ahmedabad, Country : India.

`A^2+B^2+C^2=D^2` In Ambalal Cuboid Theory- an

If `A=a` (given) where `a` is any member of `[:N,Z,Q,R]`

`B=n` (given) where `n` is any member of `[N]`

Then `C=(an)/(n-a)`


& `D=(an)/(n-a)+n-a` means `D=C+(n-a)`


Results `A^2+B^2+C^2+D^2` become true for any value

of `a E [N,Z,Q,R]` & `b E N`.


Proof `D^2-C^2=[(an)/(n-a)+n-a]^2-[(an)/(n-a)]^2`

`=((2an)/(n-a)+n-a)(n-a)`

`=2an+(n-a)^2`

`=2an+n^2-2na+a^2`

`=n^2+a^2`

`∴ D^2-C^2=B^2+A^2`

`∴ A^2+B^2+C^2=D^2`

Example 1:

Let `A=51,B=53,C=(51*53)/(53-51)=2703/2,D=2703/2+(53-5)`

`C=1351.5, D=1353.5`

Then `A^2+B^2+C^2=D^2`

`∴ (51)^2+(53)^2+(1351.5)^2=(1353.5)^2`

`∴ 2,601+2,809+18,26,552.25=18,31,962.25`

Example 2:

`A=-47, B=-51, C=((-47)(-51))/((-51)-(-47))+2397/(-4)=-599.25`

`D=-599.25+[-51-(-47)]`

`=-599.25-4`

`=-603.25` >
Then `A^2+B^2+C^2=D^2`

`∴ (-47)^2+(-51)^2+(-599.25)^2=(-603.25)^2`

`∴ 2,209+2,601+3,59,100.5625=3,63,910.5625`


Example 3:

Let `A=2.2, B=1.6, C=((2.2)(1.6))/(1.6-2.2)=3.52/-0.6`


`D=3.52/-6+(1.6-2.2)=3.52/-0.6-0.6=3.52+.36/-0.6=3.88/-0.6`

Then `A^2+B^2+C^2=D^2`

`∴ (2.2)^2+(1.6)^2+(-3.52/0.6)^2``(-3.88/0.6)^2`

`∴ 4.84(.36/.36)+2.56(.36/.36)+12.3904/.36=15.0544/.36`

`1.7424/.36+0.9216/.36+12.3904/.36=15.0544/.36`

Example 4:

Let `A=2 / 3,B=3 /5,C=((2/3*3/2))/((3/2*2/3))=((2/5))/((-1/15))=-6`


`D=-6+(⅗-2/3)=-6-1/15=-91/15`

Then `A^2+B^2+C^2=D^2`

`(2/3)^2+(3/5)^2+(-6)^2+(-91/15)^2`

`4/9(25/25)+9/25(9/9)+36(225/225)=8281/225`

`100/225+81/225+8100/225=8281/225`

Now for satisfaction take `A=aE [N,Z,Q,R] B= nEN,C=an/n-a` & `D=(an)/(n-a)+n-a`

And put in formula `A^2+B^2+C^2=D^2` to get successful result.

For any doubt and additional details please contact me via Email : ambalalparmar40@yahoo.com

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