So I'm wondering how to get the point that is equidistant between two other points. The catch is that I want it so that the equidistant point is on a specific line going at a specified angle.
Only one of these points also lies on this line while the other point is on another line that is perpendicular to that first line. Take a look at the image that is linked:
Essentially, I want to know the formula that gives me the closest equidistant point between point $A$ and point $B$ .The equidistant point must be on line $AC$ .
Let's assume I know the coordinates of points $A$ , $B$ and $C$ . I also know the length of line $AC$ and $BC$ (they aren't necessarily the same). Finally, I know the angle of the line that is defined by points $A$ and $C$ .
I know there are only two possible points that are equidistant between these two (assuming the line that is $AC$ goes on infinitely). I want the one that is closest, which has to be between points $A$ and $C$ .
The distance of line $BC$ is always less than the distance of $AC$ .
Does that all make sense?
There exists a line $L$ for which any point on the line $L$ is equidistant from point A and point B.
The line $L$ is perpendicular to the line AB and passes through the midpoint between A and B.
$AB_{slope}$ = $\frac{(A_y-B_y)}{(A_x-B_x)}$
$A_y=AB_{slope}\cdot$$A_x+b$ where $b$ is the y-intercept.
The midpoint can be found by $(\frac{A_x+B_x}{2},\frac{A_y+B_y}{2})$
As previously stated, the perpendicular line also passes through the midpoint of AB. The slope of the perpendicular line is the negative reciprocal of the $AB_{slope}$.
Therefore, the slope of line $L = \frac{-1}{AB_{slope}}$
Use the coordinates from the midpoint to solve for the y-intercept of line $L$.
The process is repeated to find the equation for line $AC$.
The point that is equidistant to A and B falls on the intersection of lines $L$ and $AC$.