How can I take a medial triangle as the new frame of reference?

Question

We have a reference triangle $ABC$. In the plane of $ABC$, a point $M$ can be written $M=x.A+y.B+z.C$.

But I would like to take the medial triangle $A'B'C'$ of $ABC$ as the new reference triangle. Do you know how I can find the transformation formulas that give the matrix change frame of reference ? In other word, how can I express $M=x.A+y.B+z.C$ in terms of $A',B',C'$ ?

Many thanks for any help.

Answer 1

Suppose that $$A'=\frac{A+B}{2},\ B'=\frac{B+C}{2},\ C'=\frac{A+C}{2}.$$ This gives $$A=A'-B'+C',$$ $$B=B'-C'+A',$$ $$C=B'+C'-A'.$$ Then, \begin{align} M&=xA+yB+zC\\ &=x(A'-B'+C')+y(B'-C'+A')+z(B'+C'-A')\\ &=(x+y-z)A'+(y-x+z)B'+(x-y+z)C'. \end{align}