{"raw_statement":[{"iden":"statement","content":"You have a triangle with three points given in coordinates. Given this information, figure out the triangle's area. The points can be given in any order.\n\nTo calculate the area of a triangle, you can use Heron's Formula, which is described below:\n\nDefine $A$, $B$, and $C$ to be the lengths of the three sides of the triangle.\n\nDefine $S$ to be equal to $frac(A + B + C, 2)$.\n\nThen, the area of the triangle is calculated as $sqrt(S (S -A) (S -B) (S -C))$.\n\nThe input consists of three lines, each consisting of two space-separated integers $x$ and $y$: the coordinates of each point of the triangle.\n\nOutput a single decimal number $a$: the area of the triangle, using Heron's Formula as described above. Do not round your answer.\n\nThis triangle corresponds to the first example case:\n\n"},{"iden":"input","content":"The input consists of three lines, each consisting of two space-separated integers $x$ and $y$: the coordinates of each point of the triangle."},{"iden":"output","content":"Output a single decimal number $a$: the area of the triangle, using Heron's Formula as described above. Do not round your answer."},{"iden":"examples","content":"Input1 1\n5 4\n2 8\nOutput12.5\nInput0 0\n8 0\n4 8\nOutput32.0\nInput-1 -3\n2 7\n9 -11\nOutput62.0\n"},{"iden":"note","content":"This triangle corresponds to the first example case:  "}],"translated_statement":null,"sample_group":[],"show_order":[],"formal_statement":"**Definitions**  \nLet $ P_1 = (x_1, y_1) $, $ P_2 = (x_2, y_2) $, $ P_3 = (x_3, y_3) \\in \\mathbb{R}^2 $ be the coordinates of the three vertices of the triangle.\n\n**Constraints**  \nThe coordinates are given as real numbers (integers in input, but treated as reals for computation).\n\n**Objective**  \nCompute the area of the triangle using Heron’s Formula:\n\n1. Compute side lengths:  \n   $$\n   A = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}, \\quad\n   B = \\sqrt{(x_3 - x_2)^2 + (y_3 - y_2)^2}, \\quad\n   C = \\sqrt{(x_1 - x_3)^2 + (y_1 - y_3)^2}\n   $$\n\n2. Compute semi-perimeter:  \n   $$\n   S = \\frac{A + B + C}{2}\n   $$\n\n3. Compute area:  \n   $$\n   \\text{Area} = \\sqrt{S(S - A)(S - B)(S - C)}\n   $$","simple_statement":"Given three points in a coordinate plane, calculate the area of the triangle they form using Heron’s Formula. Do not round the result.","has_page_source":false}