# Area of a triangle with vertices on a lattice

For a vector, let denote a vector perpendicular to with the same modulus - for example, . Then, the area of a triangle ABC is Fix the points ABC at your choice. Then: 1) Calculate the area of the triangle by means of the previous expression. 2) Calculate the area of the triangle with Pick's theorem, which states that the area of a triangle with vertices on a lattice is where denotes the amount of points inside the triangle, and denotes the number of points on the sides of the triangle. 3) Find the base and the height of the triangle using trigonometry, then calculate the area. Can you find a proof for the first expression of the area?