We can find G, H, and C are on the same line. This line called Euler line.

Can you find the relationship G,H,and C? length? Ratio?

When the center of I is on the same line? Here is GSP file.

We can find the ratio of HG:GC is constant.

The ratio is HG:GC=1:2

Why? Connect a vertex (mark A) and H, the midpoint opposite side of the vertex (mark M) and C ,and AM.

Can you discover something to relate with the ratio of HG:GC?

When the original triangle is isoceles triangle, the center of I is on the Euler line. Then what is the line the 4 centers on?

Here is GSP file.

When the original triangle is equilateral triangle, the centers gather the center point of the triangle.

Connect a vertex (mark A) and H, the midpoint opposite side of the vertex (mark M) and C,and AM. Can you discover something to relate with the ratio of GH:GC?

We can discuss about the relationship between triangles AGH and MGC. through the observation various shape of original triangle we notice AH and CM are parallel. Why?

AH through the perpendicular line of opposite side to vertex A and MC through the perpendicular bisector of the side, therefore alternative angles are equal, then AH//CM

Because of alternate angles of parallel line

HAG=COG and GHA =GCH

Because two vertexes angles of each triangle are equal , AGE and MGC are similar.

By the way , G is centroid so AG:GH=2:1

Therefore HG:GC=2:1

When original triangle is isosceles triangle, C,G,H,and I are on a line. what is the line? This is a median AM perpendicular . It means, as original triangle is isoscales triangle, the median AM consist with the perpendicular bisector of the base and perpendicular segment from a vertex . Also I is on the perpendicular bisector of the base.

1. Prove a median of the base of isosceles triangle closs the base the vertical.

2. Prove a perpendicular segment from the vertex of the isoscales triangle bisect the base.

3. Prove a the perpendicular bisector of the base of isoscales triangle pass the vertex.

4. Explain ,as original triangle is isoscales triangle, why is I on the perdicular bisector of the base.

The relationship between C,G,and H is the material using the feature of centroid.

As isoscales triangle, the rerationships of C,G,H,and I are connected the proof . In Japan, Children learn about the proof of isoscales triangle 2nd grade of junior high school(13 or 14 years old) . This material is motivation for proof.