Find if a point is in a circle I am coding a video game, but I am not so good at the math.  I am hoping for some help here:
Given:


*

*$X, Y$ that is the center of the Circle

*$R$ that is the radius of the Circle

*$X_1, Y_1$ that may or may not be in the circle.


The idea is that I have a tower that will shoot at a bad guy when it comes in range.  The bad guy has an $X_1, Y_1$ coordinate that will continually update.  
I need an equation I can run to see if the bad guy is in range.
 A: The circle is defined by all points $(x_1, y_1)$ satisfying $(x_1-x)^{2} + (y_1-y)^{2} \le R^{2}$.  That is your logical condition to test if someone is in the circle.
A: $(X-X_1)^2+(Y-Y_1)^2\leq R^2$ is true if and only if the bad guy is in range, since the distance between $(X,Y)$ and $(X_1,Y_1)$ is $\sqrt{(X-X_1)^2+(Y-Y_1)^2}$.
A: You need to check the distance $d\leq \sqrt(Xp-Xc)^2+(Yp-Yc)^2$  from the center to the bad guy and since it has square root it's simpler to test $d^2\leq (Xp-Xc)^2+(Yp-Yc)^2\leq R^2$
$Xc$  and  $Yc$ are 0 since they are in the center of circle the bad guy is in range if $(Xp)^2+(Yp)^2\leq R^2$
In some Java code it's look like this
public static Point[] internalPoints(Point[] points, double radius) {

    int countPoints = 0;
    for (int i = 0; i < points.length; i++) {
        double xp = points[i].getX();
        double yp = points[i].getY();
        // points are inside the circle if d^2 <= r^2
        // d^2 = (Xp-Xc)^2 + (Yp-Yc)^2
        // Xp and Yp is the point that should be checked
        // Xc and Xc is the point center (orgin)
        // Xc and Yc are 0 you end up with d^2 = (Xp)^2 + (Yp)^2
        if (xp * xp + yp * yp <= radius * radius) {
            countPoints++;
        }
    }
    int companionVar = 0;
    Point[] pointsInside = new Point[countPoints];
    for (int j = 0; j < countPoints; j++) {
        pointsInside[companionVar] = points[j];
        companionVar++;
    }
    return pointsInside;

}

