Equation for Perpendicular Bisector of Two Points
Theorem
Let $\tuple {x_1, y_1}$ and $\tuple {y_1, y_2}$ be two points in the cartesian plane.
Let $L$ be the perpendicular bisector of the straight line through $z_1$ and $z_2$ in the complex plane.
$L$ can be expressed by the equation:
- $y - \dfrac {y_1 + y_2} 2 = \dfrac {x_1 - x_2} {y_2 - y_1} \paren {x - \dfrac {x_1 + x_2} 2}$
Proof
Let $M$ be the straight line passing through $z_1$ and $z_2$.
Let $Q$ be the midpoint of $M$.
We have that:
- $Q = \tuple {\dfrac {x_1 + x_2} 2, \dfrac {y_1 + y_2} 2}$
The slope of $M$ is $\dfrac {y_2 - y_1} {x_2 - x_1}$.
As $L$ is perpendicular to the $M$, its slope is $\dfrac {x_1 - x_2} {y_2 - y_1}$.
Thus by Equation of Straight Line in Plane: Point-Slope Form, the equation for $L$ is:
- $y - \dfrac {y_1 + y_2} 2 = \dfrac {x_1 - x_2} {y_2 - y_1} \paren {x - \dfrac {x_1 + x_2} 2}$
$\blacksquare$
I can find no actual page on the web anywhere which gives this result explicitly, so I don't know what the "standard form" may be for this line. Hence I have not tried to simplify it, as any such "simplification" only seems to make it more complicated and less intuitive.
You can help Proof Wiki by redesigning it.
To discuss this proof in more detail, feel free to use the talk page.
If you are able to fix this proof, then when you have done so you can remove this instance of {{Improve}} from the code.
If you would welcome a second opinion as to whether your improvement is correct, add an invitation to {{Proofread}} the page (see the proofread template for usage).
