Problem with JavaNullException

The problem happens at this point as you comment:

if (PrimerPuntoTriangulo.equals(SegundoPuntoTriangulo) && SegundoPuntoTriangulo.equals(TercerPuntoTriangulo)) {

The problem is that the 3 variables have a null value, therefore you can not call the equals() method in a variable with null value, so the error NullPointerException ,

you must initialize the variables:

   triangulo PrimerPuntoTriangulo = new triangulo();
   ...
   triangulo SegundoPuntoTriangulo = new triangulo();
   ...
   triangulo TercerPuntoTriangulo = new triangulo();

the same for the variables:

 punto c1 = new punto();
 ...
 punto c2 = new punto();
 ...
 punto c3 = new punto();
 ...

Update: Actually in this exercise you only need 1 triangle and its 3 points, plus you do not need to compare objects if you do not compare the properties of the objects, in this case the points. Also, avoid defining variables as static since when instantiating and defining a value this will remain in all classes.

This would be an example of what you need:

class punto {
    int x;
    int y;
}

class triangulo {
     punto a;
     punto b;
     punto c;
}

public class Ejercicio {


 public static void main(String[] args) {

        Scanner sc = new Scanner(System.in);
        punto c1 = new punto();
        System.out.println("Punto1, Ingrese la variable x");
        c1.x = sc.nextInt();
        sc.nextLine();
        System.out.println("Punto1, Ingrese la variable y");
        c1.y = sc.nextInt();
        sc.nextLine();
        punto c2 = new punto();
        System.out.println("Punto2, Ingrese la variable x");
        c2.x = sc.nextInt();
        sc.nextLine();
        System.out.println("Punto2, Ingrese la variable y");
        c2.y = sc.nextInt();
        sc.nextLine();
        punto c3 = new punto();
        System.out.println("Punto3, Ingrese la variable x");
        c3.x = sc.nextInt();
        sc.nextLine();
        System.out.println("Punto3, Ingrese la variable y");
        c3.y = sc.nextInt();
        sc.nextLine();        

        if (((c1.x == c2.x) && (c1.y == c2.y) && (c2.x == c3.x) && (c2.y == c3.y))){
           System.out.println("El triangulo es Equilatero");
        }else if (((c1.x == c2.x) && (c1.y == c2.y) || (c1.x == c3.x) && (c1.y == c3.y) || (c2.x == c3.x) && (c2.y == c3.y))){
            System.out.println("El triangulo es Isoceles");
        } else {
           System.out.println("El triangulo es Escaleno");
        }

     }

}

When creating the new instance of the objects, do not declare them null: punto c1 = new punto(); equal in triangle triangle PrimerPuntoTriangulo= new triangulo();

You have poor validation of the objects, you are validating if the objects are the same, they will not be the same, you should validate by the points in which you set the information in each object.