I am asking for the coordinates of PointM & PointN. The problem is displayed as below figure.

Known:

1. C1 is a circle, and C2 is also a circle.
2. EC is the radius of C1 & C2.
3. PointE (Ex,Ey), PointC (Cx, Cy), length of EC

Calculated:
PointF(Fx, Fy),
sEC (slope of line EC),
bEC (y-intercept of line EC),
fMN (slope of line MN)
bMN(y-intercept of line MN)
PointM(x,y) when PointE & PointC are in the horizontal position
PointN(x,y) when PointE & PointC are in the vertical position

I have tried many options to solve the PointM & PointN coordinates when they are in general position, but have not worked so far.

Can anyone help? Java/C/C++/C#/VB/Python/pseudocode is fine.
Many Thanks!