Two lines intersection
Line 1 Equation
x + y + = 0
2 points
(x1,y1) ( ) (x2,y2) ( )
Slope, Point
(xp,yp) ( ) Slope = Angle =
Line 2 Equation
x + y + = 0
2 points
(x1,y1) ( ) (x2,y2) ( )
Slope, Point
(xp,yp) ( ) Slope = Angle =
Angle between lines
Intersection point
x =
y =
Angle
Bisector
lines
Equations
Slopes
Angles
Degree Radian    
line geometry Inclined lines
Two lines intersection
Lines given by the equations:
y = m1x + a
y = m2x + b
Where in vector notation:
A = m1i - j
B = m2i − j
The intersection point is determined by solving the values of x and y from the two lines equations by Cramer's rule or by direct substitution:
If     m2 − m1 = 0     then both lines are parallel.
The angle between two lines in the range   0 < θ < π/2   is:
Angle between two lines

The angle between the two lines can be calculated by vector dot product method:       A · B = |A||B|cos θ
Note: the ± sign stands for the two possible angles between two lines that are complementary to 180 degrees.
If the two lines are given by the equations:
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0
The intersection point is determined by solving the values of x and y from the two lines equations:
If     a1b2 − a2b1 = 0     then both lines are parallel.
If both lines are each given by two points, first line points:     (x1 , y1) , (x2 , y2) and the second line is given by two points:
(x3 , y3) , (x4 , y4)
The intersection point (x , y) is found by the equation:
Example: find the intersection point and the angle between the lines:
x − 2y + 3 = 0
3x + 4y + 1 = 0
Solving the lines equations for x and y by Cramer's rule.
And the intersection point is at (− 1.4 , 0.8)
The angle between the lines is:
If the two solutions are added then:   63.43 + 116.57 = 180 as we expected.
Two lines angle bisector
Two lines bisector angle
Lines given by the equations:
y = m1x + a
y = m2x + b
The angle of the lines angle bisector from the x axis is:
The equation of the angle bisector line is:
First line:
A second angle bisector exists at a right angle from the first line.
Second line:
Where:
mb can be expressed by the slopes m1 and m2 of the lines as:
The ± sign stands for the two angle bisectors possible between two lines (complimantery to 180 degree).
Angle bisector line equation expressed by m1 and m2 is:
y − y0 = mb(x − x0)
If the lines are given by the equations:
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0
The distance (d) of any common point (x,y) on the angle bisector between two lines are the same (two similar triangles).
distance of point (x,y) from line (1)
distance of point (x,y) from line (2)
And the equation of the lines angle bisector is:
This equation can be written as:
(a1 − φ a2)x + (b1 − φ b2)y + (c1 − φ c2) = 0
Where:
Note: The Plus and minus sign is used to find the two possible angle bisectors lines equation which are 90 degrees apart.
Example: given two lines:
3x − 4y + 2 = 0
5x + 12y + 1 = 0
find the angle between the lines and the equation of the angle bisector between the two lines.
The angle between the lines is found by vector dot product method.
We can write the lines general direction by vector notation as:
L1 = a1i + b1j       and       L2 = a2i + b2j
The dot product of this two vectors are related to the angle by the formula:       L1 · L2 = |L1||L2|cos θ
Where:
L1 · L2 = a1a2 + b1b2
Then the two possible angles are:
In order to find the angle bisector line equation we use the distance equations:
And the two lines angle bisector lines equation are:
13(3x − 4y +2) = 5(5x +12y +1)
14x − 112y + 21 = 0
And the second line:
− 13(3x − 4y +2) = 5(5x +12y +1)
64x + 8y + 31 = 0