Denote the length of DM as `n ` (an integer), the length of DA as `a , ` and DC as `b . ` Then the given conditions become
`sqrt ( n^2 + b^2 ) - sqrt ( n^2 + a^2 ) = 1 , ` `sqrt ( n^2 + a^2 + b^2 ) - sqrt ( n^2 + b^2 ) = 1 .`
The second equation gives `n^2 + b^2 = ( 1 + sqrt ( n^2 + a^2 ) )^2 ` and the sum of the two gives `n^2 + a^2 + b^2 = ( 2 + sqrt ( n^2 + a^2 ) )^2 .`
Subtract the last two equations and obtain `a^2 = 1 * ( 3 + sqrt ( n^2 + a^2 ) ) , ` `a^2 - 3 = sqrt ( n^2 + a^2 ) . ` It is also given that `sqrt ( n^2 + a^2 ) ` is an integer, so `a^2 ` is also an integer.
Further, `a^4 - 6a^2 + 9 = n^2 + a^2 , ` or `a^4 - 7a^2 + 9 - n^2 = 0 , ` a quadratic equation for `a^2 . ` Its discriminant `49 - 36 + 4n^2 = 13 + 4n^2 ` must be a square of an integer. Because 13 is prime, `13 = k^2 - 4n^2 = (k - 2n )(k+2n) ` implies `k - 2n = 1 ; ` in other words, 13 is a difference of two consecutive squares. This is possible only for `n = 3 , ` which implies `a^2 = 7 .`
From `a^2 ` and `n ` we can determine `b^2 = a^2 + 1 + 2sqrt (n^2+a^2) = 8+2sqrt(16)=16, ` and the volume `1/3 abn ` is `4sqrt(7).`