It seems like you are applying $f_1,f_2,f_3$ where 1,2,3 maps the entire
square to down right, up right, and down left respectively,
and your red dot is the solution to $f_1(f_2(f_3(x))) = x.$
Now, you just have to find these functions, which is rather easy and left as an exercise. Now, the composition of these 3 functions (in the order above) is:
$$f(x,y) = ((x+5)/8,(y-3)/8)$$
Now, solve $(x+5)/8 = x$, giving $x=5/7.$
Now, solve $(y-3)/8 = x$, giving $x=-3/7.$
The red dot should be $(5/7,-3/7).$
I have assumed that the corners of the original square is $(\pm 1,\pm 1)$.