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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 $T$?

Fixee
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Vakey
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  • "Find point given a line and two *angles" – Argon May 15 '12 at 01:14
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    Can't you ask the two angels? :) –  May 15 '12 at 01:14
  • What do you mean by "the angles of these two points"? – Argon May 15 '12 at 01:26
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    @Vakey Do you mean you know the angles subtended by $(x_3,y_3)$,$(x_1,y_1)$ at $(x_2,y_2)$ and $(x_3,y_3)$,$(x_2,y_2)$ at $(x_1,y_1)$? –  May 15 '12 at 01:31
  • I am trying to form a triangle. So those two points form one line. And lets say that the angle of point (x1, y1) is 45, and the angle of (x2, y2) is 45. Given this information, I want find the point (x3, y3). – Vakey May 15 '12 at 01:32
  • @Marvis I do not know what a subtend is. My mathematical knowledge is very low. – Vakey May 15 '12 at 01:35
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    Have a look at any exposition on "angle side angle"; for example http://www.mathsisfun.com/algebra/trig-solving-asa-triangles.html – Fixee May 15 '12 at 02:42
  • @Fixee Good article. Although, that tells me how to get two missing sides. What I am looking for the x,y coordinates of missing the point (p3) that will complete the triangle. – Vakey May 15 '12 at 16:30
  • @Vakey Once you have the two missing sides, you can write them as lines and find the unique point of intersection. That is your $p_3$. – Fixee May 15 '12 at 18:21
  • @Fixee Can you show me an example of how to find p3 given that you have those lines? – Vakey May 17 '12 at 13:25
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    @Vakey Have a look at http://zonalandeducation.com/mmts/intersections/intersectionOfTwoLines1/intersectionOfTwoLines1.html – Fixee May 17 '12 at 16:14
  • @Fixee Awesome. Thanks! – Vakey May 18 '12 at 12:30

1 Answers1

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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.

Argon
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  • Can you please check line functions . They should be $f(x)=\tan (\theta_1)x+b_1$ and $g(x)=\tan (\theta_2)x+b_2$. should not? – Mathlover May 15 '12 at 12:53
  • Thanks. Though I am having trouble finding x3. I used an example where x1=0, y1=4, x2=4, y2=0, θ1 = 45, θ2 = 45. Pointing this out on a graph, I have found that x3=4 and y3=4. Using your formula above I have found that arctan(θ1)−arctan(θ2) = arctan(45)−arctan(45) = 1.5-1.5 = 0. And you can't divide by zero. Is my math wrong, or is this a special case? – Vakey May 15 '12 at 13:27
  • @Mathlover Indeed it is $\tan$, my mistake. I never remember which one to use! – Argon May 15 '12 at 19:48
  • @Vakey They are not both $45^\circ$. One should be $-45 ^\circ=315 ^\circ$ – Argon May 15 '12 at 19:55
  • @Argon Thanks! That works. However, how do you get y3? – Vakey May 16 '12 at 12:59
  • @Vakey $y_3=f(x_3)$ – Argon May 16 '12 at 17:43
  • Awesome! That works. Thanks a bunch! – Vakey May 17 '12 at 13:24