[Math] what is the coordinate of the third point in a triangle

trigonometry

I have a triangle in the Cartesian planeˏ I know the coordinates of only two points of this triangle (none of these two points are on the origin (0,0))ˏ I also
know it's sides lengths and all it's angles.

How can I find the coordinates of the third unknown point in this triangle?

Thanks a lot

Best Answer

Say you know the coordinates of the points $A$ and $B$. Then $C$ must lie on the circle centred at $A$ with radius $AC$. Simultaneously, it must lie on the circle centred at $B$ with radius $BC$.

You can express each of these two facts as a quadratic equation in the variables being the coordinates of $C$. If the coordinates of $A$ and $B$ and all the side lengths are compatible, this set will have exactly two solutions. Those will be the two possible locations of $C$.

Related Solutions

[Math] Calculate third point of triangle from two points and angles

Let us fix some notations:

let $\alpha_1,\alpha_2,\alpha_3$ be "angle1,angle2,angle3" resp.
let length of $p_1p_2 = a_3$ and the other sides' lengthes names by cyclic permutation.
let $u=x_2-x_1, v=y_2-y_1$. Thus $a_3=\sqrt{u^2+v^2}$.

First of all: $\alpha_3=\pi-(\alpha_1+\alpha_2)$.

Then, using the law of sines (https://en.wikipedia.org/wiki/Law_of_sines):

$$\dfrac{a_1}{\sin \alpha_1}=\dfrac{a_2}{\sin \alpha_2}=\dfrac{a_3}{\sin \alpha_3}$$

one obtains in particular $a_2=a_3\dfrac{\sin \alpha_2}{\sin \alpha_3}$ where where $\alpha_2,\alpha_3$ and $a_3$ are known quantities.

Let us now express

the dot product $\vec{p_1p_2}.\vec{p_1p_3}=a_2 a_3 \cos \alpha_1$ and
the norm of the cross product $\|\vec{p_1p_2}\times\vec{p_1p_3}\|=a_2 a_3 \sin \alpha_1$

by using coordinates:

$$\begin{cases} u(x_3-x_1)+v(y_3-y_1)&=&a_2 a_3 \cos \alpha_1\\ u(y_3-y_2)-v(x_3-x_2)&=&a_2 a_3 \sin \alpha_1\end{cases}$$

One obtains a linear system of 2 equations with the two unknowns $x_3$ and $y_3$ ; the solution of this system is without difficulty.

Here is a complete Matlab program with explicit formulas for $x_3$ and $y_3$:

x1=0;y1=0;x2=6;y2=0; % initial data
alp1=2*pi/3;alp2=pi/6; % initial data
u=x2-x1;v=y2-y1;a3=sqrt(u^2+v^2);
alp3=pi-alp1-alp2;
a2=a3*sin(alp2)/sin(alp3);
RHS1=x1*u+y1*v+a2*a3*cos(alp1);
RHS2=y2*u-x2*v-a2*a3*sin(alp1);
x3=(1/a3^2)*(u*RHS1-v*RHS2);
y3=(1/a3^2)*(v*RHS1+u*RHS2);

Best Answer

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[Math] Calculate third point of triangle from two points and angles

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