Coordinate geometry or Analytic geometry
     Points, Lines and Planes in Three-Dimensional (3D) Space
      Position of two lines in a three-dimensional space
         Coplanar lines and skew lines
      Line and plane in a three-dimensional space
      Two planes in a three-dimensional space
         The line of intersection of two planes
      Orthogonality of line and plane
      Orthogonality of two planes
      Orthogonal projection of a point onto a plane
         The distance from a point to a plane
      Angle between a line and a plane, and angle between two planes
     Plane Analytic Geometry
      Coordinate plane, points, line segments and lines
         The distance formula
         The midpoint formula
      Dividing a line segment in a given ratio
      Area of a triangle
         The coordinates of the centroid of a triangle
Position of two lines in a three-dimensional space
Exactly one line passes through two different points in a space. 
Three points not on the same line define a plane.
A plane is also defined by:
    - a line and a point not on the line;      - a point and a line, which is normal/perpendicular to the plane; 
    - two lines which intersect;                 - two lines which are parallel. 
Two lines are parallel if they do not intersect and they are coplanar (lie on the same plane). 
Two lines which do not intersect but which are not coplanar are called skew lines.
Skew lines do not lie in a single plane together.
 
The relations between points, lines and planes are represented by sides, edges, diagonals and vertices of the rectangular parallelepiped (cuboid) shown in the pictures below.
   
Line and plane in a three-dimensional space
A line l intersects a plane P at a point A, as shows the left picture.
A line is defined to be parallel to a plane if the line and the plane are disjoint (empty intersection). That is, there is a line in the plane which is parallel to the given line, as shows the right picture.
The line k below lies in the plane P1 and P2 .
 
Two planes in a three-dimensional space
Two distinct planes either are disjoint (empty intersection) or intersect in a line.
Two disjoint planes are called parallel.
   
Orthogonality of line and plane
A line is perpendicular to a plane if it is perpendicular to every line in the plane that passes through its intersection point.
The plane P2 is perpendicular to the plane P1 if there is a line in P2 which is perpendicular to the plane P1 .
   
Orthogonal projection of a point onto a plane
Orthogonal projection of a point P onto a plane is the intersection point P' of the perpendicular drawn from P to the plane of projection.
Orthogonal projection of a line segment is a line segment or point depending of the position of the line segment in relation to the plane of projection.
The projection will be a point if the line segment is perpendicular to the plane of projection.
 
The distance from a point to a plane is the distance from the point to its orthogonal projection to the plane, i.e.,  AA'  = d(A,A').
For example, if the surface of a triangle and the plane of projection are mutually perpendicular, as is shown in the picture below, then the projection of the triangle is a line segment, otherwise the projection is a triangle.
 
Angle between a line and a plane, and angle between two planes
The angle between a line and a plane is the angle between the line and its orthogonal projection to the plane. 
The angle between two planes is the angle between two lines, one lying in each plane, drawn perpendicular to the intersection of the planes at the same point, as is shown below.
 
Plane Analytic Geometry
Coordinate plane, points, line segments and lines
The distance formula
The distance between two given points in a coordinate (Cartesian) plane.
 
The midpoint formula
The point on a line segment that is equidistant from its endpoints is called the midpoint.
   
Dividing a line segment in a given ratio
A given line segment AB in a Cartesian plane can be divided by a point P in a fixed ratio, internally or externally.
If P lies between endpoints then it divides AB internally. If P lies beyond the endpoints A and B it divides the segment AB externally.
The ratio of the directed lengths   l = AP : BP
is negative in the case of the internal division since the segments AP and BP have opposite sense, while in the external division, the ratio l is positive.
   
As       l = AP : BP 
and shown triangles are similar, then
 
which, with l negative, gives
the coordinates of the point P.
 
Area of a triangle
The rectangular coordinates of three points in a coordinate plane describe a triangle. Using given coordinates we derive the formula for the area of the triangle, as is shown in the diagram below.
 
The area of the given triangle P1P2P3 equals
the area of the trapezium P1MNP3 minus the
sum of the areas of the right triangles, P1MP2 and
P2NP3, that is
PD = 1/2[(y1 - y2) + (y3 - y2)] (x3 - x1) -
- 1/2[(y1 - y2)(x2 - x1) + (y3 - y2)(x3 - x2)]
which after simplifying and rearranging gives
PD=1/2[x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)]
The coordinates of the centroid of a triangle
The point of coincidence of the medians of a triangle is called the centroid.
The median is a straight line joining one vertex of a triangle to the midpoint of the opposite side and divides the triangle into two equal areas.
The centroid cuts every median in the ratio 2 : 1 from a vertex to the midpoint of the opposite side.
The coordinates of the centroid of a triangle given its three points, P1, P2 and P3 in a coordinate plane:
The centroid M(x, y), where x = (x1 + x2 + x3) / 3,    y = (y1 + y2 + y3) / 3
Coordinate geometry contents
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