When you draw the line from A through G, the intersection of two medians from the other two vertices, the line will be the last midpoint…

To explain the above axiom, we will first explain that the intersection G of two medians BG:GN = CG:GL = 2:1.

If only the case of BG:GN is explained, the same explanation will be valid for the other.

For the discussion a supplementary line is needed. Draw it from the midpoint L to AC, parallel to BN. Let K to denote the intersection of the line and AC.

As AL=LB and LK//BN, **AK=KN** (This requires a proof, two, but I’d like to skip it, because it would be visibly understandable)

Then it is derived that **CN=2NK=2KA**, as N is the midpoint of AC.

And **BN=2LK**, as the triangle ALK is just 1/2 scale of the triangle ABN.

Let’s look at LK in a different triangle, CLK, including a smallaer triangle CGN.

As CN=2NK, CK=3NK.

As G is a point on BN, LK//GN. So the triangle CGN is 2/3 scale of the triangle CLK.

So, LK:GN=3:2.

As BN=2LK, BN:LK:GN=6:3:2.

As BG=BN-GN, BG:GN=4:2=2:1.

In this way, it was proved that BG:GN=2:1 .

It’s a crazy problem of geometry.

What are you doin’ here?

Wanna be a “Einstein”? :)

When I was a child, people around me used to say I would be a greater scientist than Einstein. And 40 years later I am struggling with high-school math.

*oho* Are you now greater then Einstein? ;)

If he had been bad at geometry at school. That doesn’t mean I am good at quantum science, though.

Me, too :)

Why you do that? Hobby or is it your Job?

Sorry about my bad english… long time ago, that i have used it.

Hobby, yes. It’s my hobby to learn and share the joy of learning. When I had this problem of the gravity center, I did not see how to prove it at all. Then my teacher gave me an excellent explanation, that made me very happy.