I included the assignment as an attachment. I am trying to prove Feuerbach's Nine Point circle theorem using dilations. For the first question, I think that if if I let where Q and Q' are points on their respective circles, then I essentially have two triangles formed where side OQ has length r, radius of and side O'Q' with length r', radius of . I am not sure if my reasoning is valid but if it is, how can I use this to deduce that Q' will always be on .

I really appreciate the help.