Let the $3$ numbers be $x-y , x , x+y$
Then, $x-y +x + x+y =15$
$\implies 3x= 15, x = 5$
Now put this value into their squares. $(5-y)^2 + 25 +( 5+y)^2 = 83$ and we get $y^2 = 4 \implies y = \pm 2.$
Hence, numbers are $5, 7,3 .$
And highest part is $7$