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Explanation

Since, diagonals of rectangle bisect each other, so mid point of `(1,3)` and `(5,1)` must satisfy `y=2x+c`, i.e. `(3,2)` lies on it.

`implies2=6+c`
`impliesc=-4`
`:.` Other two vertices lies on `y=2x-4`
Let the coordinate of `B` be `(x,2x-4)`.
`:.` Slope of `AB*`Slope of `BC=-1`
`implies ((2x-4-3)/(x-1))*((2x-4-1)/(x-5))=-1`
`implies(x^(2)-6x+8)=0`
`impliesx=4,2`
`implies y=4,0`
Hence, required points are `(4,4)`, `(2,0)`