[Math] Find point given a line and two angles


Let's say I have two points $p_1=(x_1, y_1)$ and $p_2=(x_2, y_2)$, which are given as two points of a triangle $T$. And let's say I know the angles of $T$ at $p_1$ and $p_2$. How do I find the third point $p_3=(x_3, y_3)$ of $T$?

Best Answer

I believe that this is what you want. We can use basic linear algebra.

We have the two sides of the triangle. I write these sides as lines:

$$f(x)=m_1x+b_1=\frac{y_1-y_3}{x_1-x_3}x+b_1$$ $$g(x)=m_2x+b_2=\frac{y_2-y_3}{x_2-x_3}x+b_2$$

$m$ is the slope of the line. I assume that the angles you are talking about are the angles of elevation or depression of the lines $f$ and $g$. To turn an angle into slope, we use $\tan \theta$ where $\theta$ is the angle. Thus

$$f(x)=\tan (\theta_1)x+b_1$$ $$g(x)=\tan (\theta_2)x+b_2$$

To find $b_1$ and $b_2$, solve for them in $f(x_1)=y_1$ and $g(x_2)=y_2$ respectively.

$$f(x_1)=y_1=\tan (\theta_1)x_1+b_1\implies b_1=y_1-\tan (\theta_1)x_1$$ $$g(x_2)=y_2=\tan (\theta_2)x_2+b_2\implies b_2=y_2-\tan (\theta_2)x_2$$

To find $x_3$, we know this is the unique point such that $f(x_3)=g(x_3)$. Thus

$$ f(x_3)=g(x_3)=\tan (\theta_1)x_3+y_1-\tan (\theta_1)x_1= \tan (\theta_2)x_3+y_2-\tan (\theta_2)x_2 \implies x_3= \frac{\tan (\theta_1)x_1-\tan (\theta_2)x_2+y_2-y_1}{\tan (\theta_1)-\tan (\theta_2)}$$

I'm sure there are other ways to do this, but this is the first thing that comes to mind.

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