In a plane, a reflection across line AB maps point P (not on line AB) to point P prime. If the coordinates of A are (a,b), the coordinates of b are (c,d) and the coordinates of P are (x,y), then what are the coordinates of P prime in terms of a,b,c,d,x, and y?
You could also think of it this way:
In a plane with line AB, point P, and point Q (this corresponds to P prime above). The coordinates of A, B, and P are the same as above. What are the coordinates of Q such that line AB perpendicularly bisects segment PQ?