A square is inscribed in the circle x2 +y2 -10x-6y+30=0.One side of the square is parallel to the liney=x+3,then one vertex of the square is

radius of circle = 2 = half of diagonal of square

therefore perpendicular distance of center of circle(5,3) from side of square=√2

equation of one side of square : y=x+k

by distance formula,

|5+k-3|/√2=√2

k=0,-4

the two side of square are: y=x and y=x-4

solving the lines with the equation of circle, we get:

vertices: (3,3),(5,5),(5,1),(7,3)