已经知道三角形三点A(X1,Y1) B(X2,Y2) C(X3,Y3)
[vec{AB} = (X2-X1,Y2-Y1)
]
[vec{AC} = (X3-X1,Y3-Y1)
]
[||n|| = vec{AB} imes vec{AC} = |vec{AB}|cdot|vec{AB}|*Sin<vec{AB},vec{AC}>
]
[因为 |vec{AB}|*Sin<vec{AB},vec{AC}> 为三角形的高
]
[所以 S_{三角形}= frac{1}{2} | vec{AB} imes vec{AC}|
]
[=
egin{vmatrix}
X2-X1 & Y2-Y1\
X3-X1 & Y3-Y1
end{vmatrix}
]
[= (X2-X1)(Y3-Y1) * (X3-X1)(Y2-Y1)
]
[= X1Y2 + X2Y3 + X3Y1 - X1Y3 - X2Y1 - X3Y2
]