Answer:
a square is cyclic
Explanation:
You want to prove that the quadrilateral whose vertices are (1, 3), (1, -1), (3, 1), (-1, 1) is cyclic.
Midpoints
Plotting the points, we see that the first pair define one diagonal, and the second pair define the other diagonal. The midpoints of these diagonals are ...
((1, 3) +(1, -1))/2 = (1+1, 3-1)/2 = (2, 2)/2 = (1, 1)
and
((3, 1) +(-1, 1))/2 = (3-1, 1+1)/2 = (2, 2)/2 = (1, 1)
The diagonals have the same midpoint, so bisect each other.
Lengths
The length of the first diagonal is ...
(1, 3) -(1, -1) = (1-1, 3+1) = (0, 4) . . . . . . 4 units vertically
and the length of the second diagonal is ...
(3, 1) -(-1, 1) = (3+1, 1-1) = (4, 0) . . . . . . 4 units horizontally
The diagonals have the same length, and bisect each other, so the figure is a rectangle. A rectangle is a cyclic quadrilateral.
This quadrilateral is cyclic.
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Additional comment
A cyclic polygon is one whose vertices lie on a circle. This polygon is a square, so can be circumscribed by a circle as shown in the attachment.