Given the points A,B,C, defining lines BA and BC, consider two other points on these lines correspondingly: D = (1-s)B+sA, E = (1-t)B+C, with fixed s, t. On the same lines correspondingly consider also the points G = (1-x)D + xA and H = (1-x)E + xC. Vary the real number x and on the resulting variable line GH(x) consider the point I defined through I = (1-t)H+tG with fixed t. Prove that for variable x and all other data fixed, point I moves on a line (red), passing through F = (1-t)C + tA.

In the file Analogon.html we discuss the relation of point J from B and the other data.

Look at the file Trianalogon.html for an application of this property on the sides of a triangle.