§2 Basic calculations in analytic geometry
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Calculation formula 
[ distance between two points ]

on flat surface _{} in space _{}

[ The dividing line segment is the dividing point coordinates of the fixed ratio ] ( a ) ( b ) M ( x , y , z ) is the division point of A _{1 }A _{2} , the division ratio_{} 
on flat surface _{}
in space _{} λ > 0 is called inner division and outer division . When λ = 1 , M is the midpoint of A _{1 }A _{2 :}_{}_{}_{} _{}, ,_{}_{}

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Calculation formula 
[ Area of a triangle on a plane ] A _{1} , A _{2} , A _{3} form a counterclockwise loop 
S _{△} =_{} When S _{△} =0 , the three points of A _{1} , A _{2} , and A _{3} are collinear 
[ Area of a polygon on a plane ]
A _{1} , A _{2} , … , An form a _{counterclockwise} loop 
S =_{} When S = 0 , A _{1} , A _{2} , … , A _{n} n points are collinear

[ volume of a tetrahedron in space ]
right handed system_{} 
V = =_{}_{} When V = 0 , the four points M , A _{1} , A _{2} , and A _{3} are coplanar

[ Angle of the dihedral ]
M _{0} , M _{1} , A _{1} , A _{2} are four noncoplanar points in space, q is the dihedral angle A _{1}  M _{0 }M _{1}  A _{2} (0 ￡ q ￡ 180 ° )

_{} When M _{0} is the origin, the positive direction of the x axis, and the positive direction of the y axis, then_{}_{} _{} When M _{0} is the origin and the positive direction of the x axis, then_{} _{}

graphics 
Calculation formula 
[ Barycentric coordinates of mass point system ] The mass of each particle in space is m _{i} , and the coordinates are ( x _{i} , y _{i} , z _{i} ) i =1, 2, … , n 
The coordinates of the center of gravity G are_{} _{} where ∑ represents the sum of i from 1 to n 