EDIT
I'm going to have to rethink this. I'm not sure how to explain better, probably because my understanding is so fragile. Thank you for your comments and I will attempt to come back later with something better.
new here. Please pardon any question/formatting errors or missteps. I'll make edits, if necessary.
Background
I'm working on a project and am trying to find the $(x,y)$ of a third point on a triangle. I've pretty much worked out how to do this, but I come up with four potential answers:
$\left(+x,+y\right)$
$\left(+x,-y\right)$
$\left(-x,+y\right)$
$\left(-x,-y\right)$
In the below image link, I've drawn an example (pardon the poor quality). In this image, I have 5 points: $P_1$ to $P_5$, where $P_4$ is off the image in the direction of the line and $P_5$ is the point I am trying to find.
Some constants:
- This is 2 dimensional for my purposes
- The layout can fall at any angle or rotation but,
- $P_1$ is always $(0,0)$
- $P_5$ is always in line with $P_3$ and $P_4$
- $P_5$ will fall opposite of $P_3$ from $P_4$ (using the image linked, the correct option would be $P_5d$)
Question
How can I test for the "correct" one $(P_5)$ of the four answers, given that I know everything else? I'm thinking that I should be able to take the slope of the existing points and be able to calculate the answer, but I'm at a loss on where to start.
Reference threads:
- Determine the third point in right triangle only knowing the coordinates of the other two points
- How to find coordinates of 3rd vertex of a right angled triangle when everything else is known?
- Calculate coordinates of 3rd point (vertex) of a scalene triangle if angles and sides are known.
- How to find the third coordinate of a right triangle given 2 coordinates and lengths of each side
- various other threads/sites, which haven't always worked out for me
Conclusion
Thank you in advance! I can provide specific calculations if desired, but not sure how much of what to put, since I don't think anyone wants my full calc list posted. (maybe? maybe not?)
