line integration in cartesian coordinates, so is the choice of zero heading () immaterial here. Formulas resembling Heron's formula Three formulas have the same structure as Heron's formula but are expressed in terms of different variables. First, denoting the medians from sides a, b, and c respectively as ma, mb, and mc and their semi-sum as σ, we have Next, denoting the altitudes from sides a, b, and c respectively as ha, hb, and hc, and denoting the semi-sum of the reciprocals of the altitudes as we have And denoting the semi-sum of the angles' sines as , we have where D is the diameter of the circumcircle: Using Pick's theorem See Pick's theorem for a technique for finding the area of any arbitrary lattice polygon (one drawn on a grid with vertically and horizontally adjacent lattice points at equal distances, and with vertices on lattice points). The theorem states: where is the number of internal lattice points and B is the number of lattice points lying on the border of the polygon. Other area formulas Numerous other area formulas exist, such as where r is the inradius, and s is the semiperimeter (in fact, this formula holds for all tangential polygons), and where are the radii of the excircles tangent to sides a, b, c respectively. We also have and for circumdiameter D; and for angle α ≠ 90°. The area can also be expressed as In 1885, Baker gave a collection of over a hundred distinct area formulas for the triangle. These include: for circumradius (radius of the circumcircle) R, and Upper bound on the area The area T of any triangle with perimeter p satisfies with equality holding if and only if the triangle is equilateral. Other upper bounds on the area T are given by and both again holding if and only if the triangle is equilateral. Bisecting the area There are infinitely many lines that bisect the area of a triangle. Three of them are the medians, which are the only area bisectors that go through the centroid. Three other area bisectors are parallel to the triangle's sides. Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter. There can be one, two, or three of these for any given triangle. Further formulas for general Euclidean triangles The formulas in this section are true