Question: For some field $F$, given a set of $n$-dimensional input vectors $V\_{in}$ and a set of $n$-dimensional output vectors $V\_{out}$, where each of the set does not contain any duplicated elements.

Is it possible to create a corresponding linear transformation matrix $M$ such that $\vec{v}\_{out} = M \vec{v}\_{in} $ where $\vec{v}\_{in} \in V\_{in}, \vec{v}\_{out} \in V\_{out}$?

$M:F^n \rightarrow F^n$

Answer: 
You need to find the sine and/or cosine of the angles that points A and/or C have with respect to the axes. Then find the same for the acute angles within the triangle. Geometry will then tell you the angles that BC has with respect to the axes and you can find the coordinates by multiplying the length of BC by these respective values and adding any offset from C if C is not on the origin.
**Hint:**

Convert the coordinates of $A$ to polar with the pole at $C$. Then replace the modulus ($\rho=AC$) by $BC$ and add/subtract $\dfrac\pi2$ to the argument $\theta$. Convert back to Cartesian.