Trianalogon

Given triangle ABC and a number x, consider an orientation A- >B- >C and take on the sides AB, BC, CA respectively points D, E and F such that AD/AB = BE/BC = CA/CF = x. Show that:
(1) The locus of the middle H of DE is a line parallel to the side CA. Analogous statements hold for the other sides of the triangle DEF.
(2) More generally, given a number y, the locus of a point H on DE, such that DH/DE = y is a line.
(3) ABC and all the triangles DEF(x) share the same centroid I.

In the figure above varying y shows the position of the corresponding (blue) line, which is the geometric locus referred in (2) of the above proposition. This is discussed also in the file Analogon2.html .

http://users.math.uoc.gr/~pamfilos/