Task: Circles of given radius through two points
Java: import java.util.Objects;

public class Circles {
    private static class Point {
        private final double x, y;

        public Point(Double x, Double y) {
            this.x = x;
            this.y = y;
        }

        public double distanceFrom(Point other) {
            double dx = x - other.x;
            double dy = y - other.y;
            return Math.sqrt(dx * dx + dy * dy);
        }

        @Override
        public boolean equals(Object other) {
            if (this == other) return true;
            if (other == null || getClass() != other.getClass()) return false;
            Point point = (Point) other;
            return x == point.x && y == point.y;
        }

        @Override
        public String toString() {
            return String.format("(%.4f, %.4f)", x, y);
        }
    }

    private static Point[] findCircles(Point p1, Point p2, double r) {
        if (r < 0.0) throw new IllegalArgumentException("the radius can't be negative");
        if (r == 0.0 && p1 != p2) throw new IllegalArgumentException("no circles can ever be drawn");
        if (r == 0.0) return new Point[]{p1, p1};
        if (Objects.equals(p1, p2)) throw new IllegalArgumentException("an infinite number of circles can be drawn");
        double distance = p1.distanceFrom(p2);
        double diameter = 2.0 * r;
        if (distance > diameter) throw new IllegalArgumentException("the points are too far apart to draw a circle");
        Point center = new Point((p1.x + p2.x) / 2.0, (p1.y + p2.y) / 2.0);
        if (distance == diameter) return new Point[]{center, center};
        double mirrorDistance = Math.sqrt(r * r - distance * distance / 4.0);
        double dx = (p2.x - p1.x) * mirrorDistance / distance;
        double dy = (p2.y - p1.y) * mirrorDistance / distance;
        return new Point[]{
            new Point(center.x - dy, center.y + dx),
            new Point(center.x + dy, center.y - dx)
        };
    }

    public static void main(String[] args) {
        Point[] p = new Point[]{
            new Point(0.1234, 0.9876),
            new Point(0.8765, 0.2345),
            new Point(0.0000, 2.0000),
            new Point(0.0000, 0.0000)
        };
        Point[][] points = new Point[][]{
            {p[0], p[1]},
            {p[2], p[3]},
            {p[0], p[0]},
            {p[0], p[1]},
            {p[0], p[0]},
        };
        double[] radii = new double[]{2.0, 1.0, 2.0, 0.5, 0.0};
        for (int i = 0; i < radii.length; ++i) {
            Point p1 = points[i][0];
            Point p2 = points[i][1];
            double r = radii[i];
            System.out.printf("For points %s and %s with radius %f\n", p1, p2, r);
            try {
                Point[] circles = findCircles(p1, p2, r);
                Point c1 = circles[0];
                Point c2 = circles[1];
                if (Objects.equals(c1, c2)) {
                    System.out.printf("there is just one circle with center at %s\n", c1);
                } else {
                    System.out.printf("there are two circles with centers at %s and %s\n", c1, c2);
                }
            } catch (IllegalArgumentException ex) {
                System.out.println(ex.getMessage());
            }
            System.out.println();
        }
    }
}

VB: Public Sub circles()
    tests = [{0.1234, 0.9876, 0.8765, 0.2345, 2.0; 0.0000, 2.0000, 0.0000, 0.0000, 1.0; 0.1234, 0.9876, 0.1234, 0.9876, 2.0; 0.1234, 0.9876, 0.8765, 0.2345, 0.5; 0.1234, 0.9876, 0.1234, 0.9876, 0.0}]
    For i = 1 To UBound(tests)
        x1 = tests(i, 1)
        y1 = tests(i, 2)
        x2 = tests(i, 3)
        y2 = tests(i, 4)
        R = tests(i, 5)
        xd = x2 - x1
        yd = y1 - y2
        s2 = xd * xd + yd * yd
        sep = Sqr(s2)
        xh = (x1 + x2) / 2
        yh = (y1 + y2) / 2
        Dim txt As String
        If sep = 0 Then
            txt = "same points/" & IIf(R = 0, "radius is zero", "infinite solutions")
        Else
            If sep = 2 * R Then
                txt = "opposite ends of diameter with centre " & xh & ", " & yh & "."
            Else
                If sep > 2 * R Then
                    txt = "too far apart " & sep & " > " & 2 * R
                Else
                    md = Sqr(R * R - s2 / 4)
                    xs = md * xd / sep
                    ys = md * yd / sep
                    txt = "{" & Format(xh + ys, "0.0000") & ", " & Format(yh + xs, "0.0000") & _
                    "} and {" & Format(xh - ys, "0.0000") & ", " & Format(yh - xs, "0.0000") & "}"
                End If
            End If
        End If
        Debug.Print "points " & "{" & x1 & ", " & y1 & "}" & ", " & "{" & x2 & ", " & y2 & "}" & " with radius " & R & " ==> " & txt
    Next i
End Sub