A Bisector

Perpendicular bisector

The term "perpendicular bisector" refers to a line or a segment that is both perpendicular to another line or segment and passes through its midpoint. In geometric terms:

1. Perpendicular: The perpendicular bisector forms a right angle (90 degrees) with the line or segment it bisects.

2. Bisector: It divides the line or segment into two equal parts, with the point of intersection being the midpoint.

Equation:

\[\frac {y_1-y_2}{x_1-x_2}\ne \frac {y_2-y_3}{x_2-x_3}\]

The perpendicular bisector of the segment\(A_1A_2\)if \(A_1(x_1,y_1)\), \(A_2(x_2,y_2)\)is perpendicular to the segment so its slope equals to \(-\frac 1a\); a is the slope of the line that passes through the \(A_1\)and \(A_2\). this bisector passes through the middle of the segment, which is the point \((\frac {x_1+x_2}2,\frac {y_1+y_2}2)\)

so the equation of the perpendicular bisect is:

\[y-\frac{y_1+y_2}{2}=-\frac {x_1-x_2}{y_1-y_2}\left [ x-\frac {x_1+x_2}2\right ]\]

or:

\[y=-\frac {x_1-x_2}{y_1-y_2}\left [ x+\frac {x_1+x_2}2\right ]-\frac{y_1+y_2}{2}\]

\[y=\frac {x_2-x_1}{y_1-y_2} x+\frac {x_2-x_1}{y_1-y_2}\frac {x_1+x_2}2-\frac{y_1+y_2}{2}\]

\[y=\frac {x_2-x_1}{y_1-y_2} x+\frac {(x_2-x_1)(x_2+x_1)+(y_2-y_1)(y_2+y_1)}{2(y_1-y_2)}\]

The point where two lines intersect

Two line:

\(\Delta_1\); \(y=a_1x+b_1\)

\(\Delta_2\); \(y=a_2x+b_2\)

\[a_1x+b_1=a_2x+b_2\]

\[-(a_2-a_1)x=b_2-b_1\]

\[x_0=-\frac {b_2-b_1}{a_2-a_1}\]

\[y_0=-a_1\frac {b_2-b_1}{a_2-a_1}+b_1\]

\[y_0=b_1-\frac {a_1(b_2-b_1)}{a_2-a_1}\]

\[y_0=\frac {b_1(a_2-a_1)-a_1(b_2-b_1)}{a_2-a_1}\]

\[A_0(-\frac {b_2-b_1}{a_2-a_1},\frac {b_1(a_2-a_1)-a_1(b_2-b_1)}{a_2-a_1})\]