2
$\begingroup$

I simply have a triangle.. say abc . Given coordinates of a & b. and the length of ac and bc..

I can calculate the length between ab via the distance square rule.

$D = \sqrt{(X_2 - X_1)^2 + (Y_2 - Y_1)^2} $

I have tried to use the distance rule to compute the third vertex, but it has a lot of steps.. I wonder if there is another method.. that yields the two possible solutions for the vertex

  • 0
    What are the X and Y? I don't quite understand what you're asking here... – 2017-02-22
  • 0
    X and Y are just place holders for the coordinates of any point in the Cartesian plane.. and D is the distance between them – 2017-02-22

2 Answers 2

3

Let $A = (x_A, y_A)$ and $B = (x_B,y_B)$ the known vertices of your triangle. Let's call $d_{AB}$, $d_{BC}$ and $d_{CA}$ the lengths of each side.

  1. Translate your points subtracting $x_A$ and $y_A$ so that $A$ corresponds with the origin. That is:

$$A' = (0, 0), B' = (x_B-x_A, y_B-y_A ) = (x_B', y_B').$$

  1. Rotate $B'$ so that it lies on the $x$-axis. This can be done without knowing the angle, indeed:

$$A'' = (0,0), B'' = (d_{AB}, 0).$$

Anyway, the value of the rotation angle is important for the next steps. In particular it is $$\theta = \arctan2\left(y_B-y_A,x_B-x_A\right),$$

where $\arctan2(\cdot, \cdot)$ is defined in details here.

  1. At this point, it is easy to find $C''$. Notice that there are two solutions, since the point $C''$ can be placed above or below the side $AB$.

$$x_C'' = \frac{d_{AB}^2+d_{AC}^2-d_{BC}^2}{2d_{AB}},$$

and

$$y_C'' = \pm\frac{\sqrt{(d_{AB}+d_{AC}+d_{BC})(d_{AB}+d_{AC}-d_{BC})(d_{AB}-d_{AC}+d_{BC})(-d_{AB}+d_{AC}+d_{BC})}}{2d_{AB}}.$$

  1. Now, rotate back your point $C''$ using $-\theta$ (see step 2), thus obtaining $C'$.
  2. Finally, translate $C'$ by adding $x_A$ and $y_A$ to the components in order to obtain $C$.
  • 2
    I was astonished .. Thanks – 2017-02-22
  • 0
    @ShadyAtef you are welcome – 2017-02-22
  • 0
    It would be nice if you also provide the way to rotate C'' in step 4. – 2018-04-24
  • 0
    This was a massive help, thank you. A few post-implementation notes: In step 2, calculate `θ = arctan2(B'y, B'x)` as it's simpler. Thus calculated, θ is the angle from the x-axis to B', so it is actually the *opposite* of the triangle's rotation angle as stated. So *don't* negate θ for step 4. Step 4, to counter-rotate your two C'' points [call them C1'' & C2''] back to C1' and C2' do: `C1'x = C''x * cos(θ) - C1''y * sin(θ)`, `C1'y = C''x * sin(θ) + C1''y * cos(0)` and `C2'x = C''x * cos(θ) - C2''y * sin(θ)`, `C2'y = C''x * sin(θ) + C2''y * cos(0)`. – 2018-07-19
1

There is much easier solution, which has 3 steps instead of 5 and doesn't require translating or rotating:

$φ_1 = \arctan2(B_y - A_y, B_x - A_x)$

$φ_2 = \arccos\left(\dfrac{l_1^2 + l_3^2 - l_2^2}{2\cdot l_1\cdot l_3}\right) $

$C = A + l_1\cdot[\cos(φ_1±φ_2)$; $\sin(φ_1±φ_2)]$

Where $A_x$, $A_y$, $B_x$, $B_y$ are your given coordinates and $l_1$, $l_2$, $l_3$ are lengths of $AC$, $BC$ and $AB$ respectively (see the image). Note that there is $±$ sign, because you can build 2 triangles which will satisfy your problem.

Problem legend

This answer may be a bit late, but I hope other people who face this problem will find my solution useful.