Lets assume I have a triangle $(p0, p1, p2)$ with $(p1 - p0)$ the longest edge. I am looking to find the point $q$ on the edge $(p1 - p0)$ such that $dot(p2 - q, p1 - p0) = 0$. That is to say; the following triangles will be right triangles: $(p0, q, p2)$ $(q, p1, p2)$.
How can I find $q$?
On
If you know the values of p0,p1 and p2, then it's just applying Pythagora's theorem to the two triangles (p0,p2,q) and (p1,p2,q). Assume that point 'q' is at a distance of some unknown say 'y' from p1. So the point will be at a distance of (length of the line segment (p0-p1) minus unknown 'y'). And assume that the line segment that divides the big triangle into two smaller right angled triangles which is (p2-q) be some other unknown say 'x'. Now apply Phythagora's theorem to both triangles (smaller triangles). Common side is (p2-q) whose length is assumed to be 'x'. After you write down the equations, just equate the values for (x^2) and the simplification will reduce the whole problem to a linear equation in 'y' which can be solved easily. Hope this helps! Sorry, if I understood your problem completely wrong.
Hint: Write $q-p_0=\lambda(p_1-p_0)=(q-p_2)+(p_2-p_0)$ and use the dot product to find $\lambda$.