What you are essentially doing is finding the vector that goes from the plane to the point which is normal to the plane, reversing it and adding two times the reverse vector to the point.
To find the perpendicular distance between a point and a plane you calculate n . p where n is the plane normal (nx,ny,nz) [A unit normal vector] and p is the point in question.
You then take your point and calculate p = p - n*2*d where n is the plane normal and d is the distance you calculated above.
If you do this for a particular value of t and leave the final reflected point arbitrary you will get your solution.
If you need a more intuitive explanation is going on I can help in that way as well.