I have a triangle defined by known vertices $p_0$, $p_1$, and $p_2$. Inside this triangle is a smaller triangle with unknown vertices $p_3$, $p_4$, and $p_5$.
Also known are the distance between some of the vertices: $d_{03}, d_{14}, d_{25}, d_{34}, d_{45}, d_{35}$
The provided image shows unknowns in red.
I have tried to solve this using a system of equations. I did the following for each of the inner-outer vertex pairs: $$ (p_{3x} - p_{0x})^2 + (p_{3y} - p_{0y}) = {d_{03}}^2 $$$$ (p_{3x} - p_{0x})(p_{3x} + p_{0x}) + (p_{3y} - p_{0y})(p_{3y} + p_{0y}) = {d_{03}}^2 $$$$ {p_{3x}}^2 - {p_{0x}}^2 + {p_{3y}}^2 - {p_{0y}}^2 = {d_{03}}^2 $$$$ {p_{3x}}^2 + {p_{3y}}^2 = {d_{03}}^2 + {p_{0x}}^2 + {p_{0y}}^2 $$ Doing this for all three inner-outer pairs provided these three equations: $$ {p_{3x}}^2 + {p_{3y}}^2 = {d_{03}}^2 + {p_{0x}}^2 + {p_{0y}}^2 $$$$ {p_{4x}}^2 + {p_{4y}}^2 = {d_{14}}^2 + {p_{1x}}^2 + {p_{1y}}^2 $$$$ {p_{5x}}^2 + {p_{5y}}^2 = {d_{25}}^2 + {p_{2x}}^2 + {p_{2y}}^2 $$
I formed three more equations by relating the inner pairs: $$ {p_{3x}}^2 + {p_{3y}}^2 - {p_{4x}}^2 - {p_{4y}}^2 = {d_{34}}^2 $$$$ {p_{4x}}^2 + {p_{4y}}^2 - {p_{5x}}^2 - {p_{5y}}^2 = {d_{45}}^2 $$$$ {p_{5x}}^2 + {p_{5y}}^2 - {p_{3x}}^2 - {p_{3y}}^2 = {d_{35}}^2 $$
This gave me six equations with six unknowns so that I can solve $Ax=b$: $$ \begin{bmatrix} 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 \\ 1 & 1 & -1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 1 & -1 & -1 \\ -1 & -1 & 0 & 0 & 1 & 1 \\ \end{bmatrix} \begin{bmatrix} {p_{3x}}^2 \\ {p_{3y}}^2 \\ {p_{4x}}^2 \\ {p_{4y}}^2 \\ {p_{5x}}^2 \\ {p_{5y}}^2 \\ \end{bmatrix}= \begin{bmatrix} {d_{03}}^2 + {p_{0x}}^2 + {p_{0y}}^2 \\ {d_{14}}^2 + {p_{1x}}^2 + {p_{1y}}^2 \\ {d_{25}}^2 + {p_{2x}}^2 + {p_{2y}}^2 \\ {d_{34}}^2 \\ {d_{45}}^2 \\ {d_{35}}^2 \\ \end{bmatrix} $$
This was my best guess at how to approach the problem. But Unfortunately this is resulting in a singular matrix, which means I must be going about this the wrong way.
I'm also aware, just by looking at the picture, that this problem should almost always result in two solutions, but my approach doesn't expose that, which is another hint that I have made a wrong turn.
I would appreciate any insight on how I can reapproach the problem.
Thanks!
This is a system of $6$ quadratic equations in $6$ unknowns which are the $x$ and $y$ coordinates of the three unknown points. This system can be solved numerically using Newton-Raphson method, and should converge to a solution within a few iterations.
I don't think it is possible to find a closed-form solution to this system.