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Let $(x_1,y_1), (x_2,y_2), (x_3,y_3)$ be variables and $(a,b),det_1, dot_1,det_2,dot_2,det_3,dot_3$ be given constants. How can I solve the following system of nonlinear equations based on constant values?

\begin{cases} x_1x_2-a(x_1+x_2)+a^2+y_1y_2-b(y_1+y_2)+b^2 = dot_1\\ x_1y_2-x_2y_1+b(x_2-x_1)+a(y_1-y_2) = det_1\\ x_2x_3-a(x_2+x_3)+a^2+y_2y_3-b(y_2+y_3)+b^2 = dot_2\\ x_2y_3-x_3y_2+b(x_3-x_2)+a(y_2-y_3) = det_2\\ x_3x_1-a(x_3+x_1)+a^2+y_3y_1-b(y_3+y_1)+b^2 = dot_3\\ x_3y_1-x_1y_3+b(x_1-x_3)+a(y_3-y_1) = det_3\\ \end{cases}

As I know it has an unlimited number of answers but I don't know how to solve it. I'm looking for answers in integer and real numbers.

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  • $\begingroup$ Yes, I tried. but I need to find general formulas for the answer. $\endgroup$ – Mandana Ghasemi May 18 at 13:06
  • $\begingroup$ 3? I see 6 equations. $\endgroup$ – Rodrigo de Azevedo May 27 at 9:13
  • $\begingroup$ $dot_k$ or $det_k$? Also what do you mean by "solve"? What form do you expect the solution to be in terms of? Why not use simpler constants such as C or K? $\endgroup$ – NoChance May 27 at 9:41
  • $\begingroup$ @NoChance My mean of "solve" is to find a general formula in order to calculate answers for x1, x2, x3, y1, y2 and y3 based on constants. $\endgroup$ – Mandana Ghasemi May 27 at 11:07
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I will just give a sketch for now how to solve it, I can fill in more details if necessary.

First define vectors $v_i = (x_i-a,y_i-b),\, i = 1,2,3$. Then we can rewrite the equations as

\begin{align} \langle v_1, v_2\rangle = a_{12},\ \langle v_2, v_3\rangle = a_{23},\ \langle v_3, v_1\rangle = a_{31},\\ \det[v_1^t\ v_2^t] = b_{12},\ \det[v_2^t\ v_3^t] = b_{23},\ \det[v_3^t\ v_1^t] = b_{31},\\ \end{align} where I changed dot's and det's to a's and b's.

Denote by $\vartheta_{ij}$ the (oriented) angle between vectors $v_i$ and $v_j$. Remember that dot product in $\mathbb R^2$ is given by formula $\|v\|\|w\|\cos\vartheta$ and that determinants above measure (signed) area of parallelogram formed by the vectors, which can also be expressed as $\|v\|\|w\|\sin\vartheta$, and thus the system becomes

\begin{align} \|v_1\|\|v_2\|\cos\vartheta_{12} = a_{12},\ \|v_2\|\|v_3\|\cos\vartheta_{23} = a_{23},\ \|v_3\|\|v_1\|\cos\vartheta_{31} = a_{31},\\ \|v_1\|\|v_2\|\sin\vartheta_{12} = b_{12},\ \|v_2\|\|v_3\|\sin\vartheta_{23} = b_{23},\ \|v_3\|\|v_1\|\sin\vartheta_{31} = b_{31}.\\ \end{align}

Then, you can find the lengths $\|v_i\|$ by looking at the system $$\|v_1\|\|v_2\| = c_{12},\ \|v_2\|\|v_3\| = c_{23},\ \|v_3\|\|v_1\| = c_{31},$$ where $c_{ij} = \sqrt{a_{ij}^2+b_{ij}^2}$ obtained by squaring the above equations and using $\sin^2t+\cos^2t = 1$. Solving it gives you $$\|v_1\|=\sqrt{\frac{c_{12}c_{31}}{c_{23}}},\ \|v_2\|=\sqrt{\frac{c_{12}c_{23}}{c_{31}}},\ \|v_3\|=\sqrt{\frac{c_{23}c_{31}}{c_{12}}}.$$

The whole thing is, from geometric perspective, obviously rotationally invariant (rotating all of the vectors won't change the angles between them or the areas), so fix some angle $\vartheta$. Then, the solution can be represented as complex numbers as

$$v_1 = \|v_1\|e^{i\vartheta},\ v_2 = \|v_2\|e^{i(\vartheta +\vartheta_{12})},\ v_3 = \|v_3\|e^{i(\vartheta-\vartheta_{31})}.$$

More explicitly, $e^{i\vartheta_{12}} = \cos\vartheta_{12} + i\sin\vartheta_{12} = \frac{a_{12}}{c_{12}}+\frac{b_{12}}{c_{12}}i$ and $e^{i\vartheta_{31}} = \cos\vartheta_{31} + i\sin\vartheta_{31} = \frac{a_{31}}{c_{31}}+\frac{b_{31}}{c_{31}}i$, so we have

\begin{align}v_1 &= \sqrt{\frac{c_{12}c_{31}}{c_{23}}}(\cos\vartheta + i\sin\vartheta),\\ v_2 &= \sqrt{\frac{c_{12}c_{23}}{c_{31}}}(\cos\vartheta + i\sin\vartheta)(\frac{a_{12}}{c_{12}}+i\frac{b_{12}}{c_{12}}),\\ v_3 &= \sqrt{\frac{c_{23}c_{31}}{c_{12}}}(\cos\vartheta + i\sin\vartheta)(\frac{a_{31}}{c_{31}}-i\frac{b_{31}}{c_{31}}),\ \vartheta\in\mathbb R.\end{align}

All you have to do now is expand and $x$'s will be the real parts, while $y$'s the imaginary parts of the above complex number representation.

Note, however, that the system is overdetermined since knowing the angles between $v_1$ and $v_2$ and $v_2$ and $v_3$ will also give you the angle between $v_1$ and $v_3$.

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  • $\begingroup$ I am bit in a rush, so the answer might contain errors. Please let me know if you notice anything so I can fix it later. Hopefully the idea gets across. $\endgroup$ – Ennar May 27 at 9:55
  • $\begingroup$ Thank you, I got you idea. However, I want to find the values of x1, x2, x3, y1, y2, and y3 based on constants. $\endgroup$ – Mandana Ghasemi May 27 at 11:04
  • $\begingroup$ something like this answer: math.stackexchange.com/questions/2894273/solving-6-equations $\endgroup$ – Mandana Ghasemi May 27 at 11:18
  • $\begingroup$ @Mandana Ghasemi, implicitly, the $x$'s and $y$'s were already described completely in terms of given constants. However, I've edited the answer to be more explicit. Do you need more details? $\endgroup$ – Ennar May 27 at 13:42
  • $\begingroup$ Thank you so much. $\endgroup$ – Mandana Ghasemi May 27 at 20:55
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Hint:

The equations can be seen as the dot and cross products of three vectors formed by the common origin $(a,b)$ and three points.

The ratios $\dfrac{\text{det}}{\text{dot}}$ give you the (tangent of the) angles they form, and the square roots of the pairwise sum-of-squares give you the products of the lengths. From the latter you can retrieve the individual lengths.

The exact positions are undetermined, only the relative directions are known. You can choose one arbitrarily and the others will follow.

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