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/
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