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