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Finding Equations of Angle Bisectors between Two Lines

Date: 08/14/2004 at 01:45:11
From: Aryan
Subject: Angle Bisector!

Is there any way or formula to find the coordinates of points which
lie on the line that bisects the angle between two given lines?

We know four points in the plane:

  P1(X1,Y1)
  P2(X2,Y2)
  P3(X3,Y3)
  P4(X4,Y4)

Line 1 passes through points P1 and P2.  Line 2 passes through points
P3 and P4.  Assume that point O (x,y) is the intersection point of the
two lines.



Date: 08/14/2004 at 09:50:43
From: Doctor Rob
Subject: Re: Angle Bisector!

Thanks for writing to Ask Dr. Math, Aryan!

Let the slope of line 1 be m1 and the slope of line 2 be m2;

   m1 = (Y2 - Y1)/(X2 - X1),
   m2 = (Y4 - Y3)/(X4 - X3).

Then the inclinations A1 and A2 of the two lines are given by

   A1 = Arctan(m1),
   A2 = Arctan(m2).

Note that if X2 - X1 = 0, then line 1 is vertical, and its inclination
A1 is Pi/2 radians or 90 degrees.  If X4 - X3 = 0, then line 2 is
vertical, and its inclination A2 is Pi/2 radians or 90 degrees.

Now notice that there are two possible angle bisectors--one of the
acute angle formed between the two given lines and one of the obtuse 
angle.  The two bisectors are perpendicular to each other.  The
inclinations B1 and B2 of the bisectors are

   B1 = (A1 + A2)/2,
   B2 = (A1 + A2 + Pi)/2  or  (A1 + A2 + 180)/2 degrees

The slopes m3 and m4 of the two angle bisectors are then

   m3 = tan(B1),
   m4 = tan(B2) = -cot(B1).

Now if neither m3 nor m4 is zero, with these two slopes and the point 
O (the intersection), you can write the equations of the angle 
bisectors,

   y = m3*(x - X) + Y,
   y = m4*(x - X) + Y.

If m3 = 0 or m4 = 0, then the angle bisector lines are vertical and
horizontal and their equations are

   y = Y,
   x = X.

Then any point with coordinates (x,y) satisfying either of the angle
bisector equations will lie on one of those two angle bisectors.

Feel free to reply if I can help further with this question.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Coordinate Plane Geometry

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