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diagram

I have 4 points: $Q, R, S, T$.

I know the following

  • Coordinates for $R$, $T$, and $S$;
  • Length of $\overline{RQ}$
  • That segment $\overline{RT} < \overline{RQ} < \overline{RS}$;
  • I need to figure out the coordinates of $Q$.

$R$, $T$, $S$ form a triangle and point $Q$ can be found on the line segment $\overline{TS}$. I need to get the coordinates for $Q$.

I have solutions for this problem, but they are all so convoluted and long I know I must be doing something wrong. I feel certain there must a simple elegant way to solve this. The best solution would be the simplest one since this needs to be programmed.

  • 1
    I suspect you can improve the question. In particular it seems you also know the length of $RQ$ to be $d=25$. A circle centered at $R$ of radius $25$ will meet the (extended) line $TS$ $0$, $1$ or $2$ times, and some of these may be between $T$ and $S$.2011-09-02
  • 1
    However because RT < RQ < RS we know that there is exactly one point on segment TS. The length of RQ is irrelevant since I need a formula not specific number.2011-09-02
  • 0
    Is the length `e` known? Otherwise, your problem's underdetermined.2011-09-02
  • 0
    @David: The key point is whether you know the length of $RQ$, as well as knowing it is more than $RT$ and less than $RS$.2011-09-02
  • 1
    BTW: I don't see a point $I$ in your diagram.2011-09-02
  • 0
    Sorry I forgot to explain I know the length of segment RQ2011-09-02
  • 0
    In that case: you want to find the intersection point of a circle of radius $d$ centered at $R$, and the line joining $T$ and $S$... which does result in a quadratic equation you need to solve. You may first want to perform a preliminary translation and rotation so that $R$ is the origin and $\overline{RS}$ lies on the horizontal axis, find the intersection point, and then undo the rotation and translation afterwards.2011-09-02

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