There is a well known formula for projection of a point b onto a line a. The sought point p can be expressed as p = x a, where x is a dot b divided by a dot a.
I have a somewhat similar situation.
I have a midpoint m of a line segment starting at b and ending at c. I draw a line through m, that is perpendicular to the line segment. This line intersects the line a at some point p. Since p is on the line a it can be expressed as p = x a . So I want to find x expressed in the terms of a, b and c. The formula has to be similar (expressed in vectors) to what I have for projection. I am trying to avoid using the cartesian coordinates of type (x1,y1), because there are 3 points involved and using coordinates would give a very complicated formula.
Step 1: Project $\frac12(\mathbf b + \mathbf c)$ onto $\mathbf c - \mathbf b,$ producing a vector $\mathbf q$.
Step 2: Project $\mathbf a$ onto $\mathbf c - \mathbf b,$ producing a vector $\mathbf r$.
Then $x = \dfrac{\lVert \mathbf q\rVert}{\lVert \mathbf r\rVert}.$