Line Geometry
(1) Line equation: y =  x +  
Two points on line: x1 y1 x2 y2
Angle (α): Intercepts:   (xi,0): (0,yi):
Midpoint: xd yd
Line length:
Point in plane: xp   yp
The point is on the line
Distance from point (xp, yp) to line (1):
Equation of line passing through (xp, yp)
and is perpendicular to line (1):
Intersection coordinates:
x:
y:
Degree Radian     








The most general equation of a line is of the form:
(1)
where A, B and C are any real number and A and B are not
both zero. If B ≠ 0 then we can divide equation (1) by B to
obtain the form:
(2)
This is the equation of a line whose slope is:
and
Equation (2) can be normalized to the general form:
(3)
the slope of the line (m) is defined in terms of the inclination
and is: (4)
Note: if the angle α is greater then 90 degree then the slope
is negative.
α (0 - 90) degree      :positive slope
α (90 - 180) degree    :negative slope
Necessary condition for two lines to be perpendicular to each other is that their slopes fulfills the condition:
m1m2 = − 1 (5)
In order to find the intersection point of two lines we have to solve the system of linear equations representing the lines.
solution by matrix:
If the determinant then intersection point exists.

Basic line of the form     y = ax + b     or     Ax + By + C = 0
Slope (m) of the line
m = a Slope m Slope m Slope m
yintercept (yi)
yi = b yi = −mxi yi = y1 − mx1 y intercept
xintercept (xi)
x intercept x intercept x intercept x intercept
tan θ
tan θ = m tan θ tan θ
Line angle (θ)
from x axis (range 0 ≤ θ < π)
θ = arctan(m) θ = arctan(m) θ = arctan(m)
Slope (M) of a line perpendicular to a given slope (m)
Slope of perpendicular line Slope of perpendicular line Slope of perpendicular line Slope of perpendicular line
Line midpointLine point
tan θ
Point (x, y) which divides the line connecting two points (x1 , y1) and (x2 , y2) in the ratio p:q
Portion of a line Portion of a line
Point (x, y) which divides the line connecting two points (x1 , y1) and (x2 , y2) externally at a ratio p:q
Portion of a line tan θ
Note: the (x,y) point is in the direction from point 1 to point 2,to get the other side extension change the point 1 with point 2 and vice versa
A point (x, y) which is located at a distance d from a point (x1 , y1) on the line
Distance from a line
Angle θ between two lines:
Angle limit
Angle between two slopes
Angle between two lines
Angle between two lines given by
Ax + By + C = 0
Dx + Ey + F = 0
Angle between two lines Angle between two lines
Lines equations
Equation of a line passing through a point (x1 , y1)
y = mx + (y1 − mx1) y − y1 = m(x − x1)
Equation of a line passing through two points (x1 , y1), (x2 , y2)
Line passing through two points Line passing through two points
Equation of a line perpendicular to a given slope m and passing through a point (xp , yp)
Equation of a line perpendicular to a given slope Equation of a line perpendicular to a given slope
Equation of a line perpendicular to a line which is defined by two points (x1 , y1) and (x2 , y2) and passing through the point (xp , yp)
Perpendicular line Perpendicular line
Equation of a line passing through the intercepts xi , yi
Intercepts point xiy = −yix + xiyi
Equation of a line passing through the point (xp ,yp) and parallel to a line which is defined by two points (x1 , y1) and (x2 , y2)
Line passing through a point Line passing through a point
Equation of a line parallel to the line Ax + By + C = 0 and at a distance d from it.
Parallel line
Equation of the midline between the lines    Ax + By + C = 0
Dx + Ey + F = 0
Midline
Equation of a line perpendicular to the line    Ax + By + C = 0
Bx − Ay + C = 0
Equation of a horizontal line
Horizontal line y = b
Equation of a vertical line
Vertical line x = a
Lines distances     y = ax + b    Ax + By + C = 0
Distance between two points (D)
Points distance
Distance between intercepts xi and yi
Intercepts distance
Distance from a line to the origin
Origin distance Origin distance
Distance from a line given by two points (x1 ,y1),(x2 , y2) to the origin
Line distance
Distance from a line to the point
(xp , yp)
Line point distance Line point distance
Distance from a line given by two points (x1 , y1) , (x2 , y2) to the point (xp , yp)
Line point distance
Distance between two parallel lines
y = ax + by = cx + d
or
Ax + By + C = 0Dx + Ey + F = 0
Parallel lines Parallel lines