This problem can be easily solved using linear algebra. At first let's assume that A coincides with A'. We may consider two coordinate systems with origin A: one has base vectors AB and AC, the other AB' and AC'. Now, let a point P have coordinates (x, y) in the first system (ABC) and P' have the same coordinates (x, y), but in the second system (AB'C'). Then the coordinates of P' in the first system can be found as the result of multiplication of the transformation matrix A and the column vector . The matrix A of the transformation T that converts the first coordinate system into the second one is obtained by "transforming each of the vectors of the standard [i.e., first] basis by T and then inserting the results into the columns of a matrix. In other words,
Let's assume that the square length is 1 in your example. Then (still assuming that A coincides with A'), vector A'B' has coordinates in the first system, and vector A'C' has coordinates . Therefore, the transformation matrix is
Let P has coordinates, say, in the first system. Then P' has coordinates
still in the first system.
If A does not coincide with A', then after the first transformation it is sufficient to have a shift by the vector AA', i.e., add the coordinates of A' in the first system (where A is the origin) to the result of matrix multiplication.
You can get a better idea if you read a good textbook on linear algebra describing linear transformations.