We start from a proposal by George A. Tsiamas in Facebook
We give a proof by using barycentric coordinates. In addition we generalize the problem for the case in which circles have different radii and finally we consider a homography from line $BC$ to line at infiinity that establishes a relationship between the ratio of the radii of the two circles and the constant direction of the lines joining the midpoints considered in the statement.
Constant direction
