I create the point G in the triangle ABC

I create the point H in the triangle ABC

I create the point C in the triangle ABC

I create the point I in the triangle ABC

Now you use and play the file of GSP I make above, by clicking here.

I found several relationships among the four point above.

1. C, G, and H are on a same line(look at this file and move any vertexe of the triangle).

Although the point I is not on the line going through C, G, and H, I comes to be on the line when the triangle is an isosceles triangle.

2. G is between C and H

3. the point I is always on the same side of the point H.

this means that if I divide the whole plane into two parts by drawinng the perpendicular line to CH, the point I is on the same area of H.

See the picture below and check this file.

4. the ratio of the distance HG and CG is 2. (See this file)

this found is easily proved.

By definition of H and C, AH and MC are parallel (both are perpendicular to BC).

Thus, the triangles AGH and MGC are similar.

Then, since the ratio of AG to MG is 2 to 1, the ratio of HG to CG is also 2 to 1.