Proving the 2:1 Ratio of a Triangle's Medians at the Centroid Using Vectors

In summary, the conversation discusses the process of proving the 2:1 ratio of a triangle's medians at the centroid using vectors. The individual is looking for clarification on how to prove the ratio and seeks help or suggestions. The conversation also mentions that adding the vectors from the centroid to the vertices results in a sum of zero, which can help prove the ratio.
  • #1
pjallen58
12
0
I am trying to figure out how to prove the 2:1 ratio of a triangle's medians at the centroid using vectors. Example if I had a triangle ABC with midpoints D of BC, E of AC and F of AB. I know G is where the medians intersect. I have seen many proofs and understand the process that proves the addition of the vectors from the centroid to the vertices are zero i.e. GA+GB+GC=0.
vectorsum_dreieck.GIF

Does this prove the 2:1 ratio? I cannot find anything explaining how to prove the actual 2:1 ratio. I am not sure if I am missing something or what. Any help or suggestions would be appreciated. Thanks.
 
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  • #2
pjallen58 said:
I am trying to figure out how to prove the 2:1 ratio of a triangle's medians at the centroid using vectors. Example if I had a triangle ABC with midpoints D of BC, E of AC and F of AB. I know G is where the medians intersect. I have seen many proofs and understand the process that proves the addition of the vectors from the centroid to the vertices are zero i.e. GA+GB+GC=0.
View attachment 14402
Does this prove the 2:1 ratio? I cannot find anything explaining how to prove the actual 2:1 ratio. I am not sure if I am missing something or what. Any help or suggestions would be appreciated. Thanks.

Hi pjallen58! :smile:

Yes, because, for example, a + 2(d) is 1/3 of the way,

and that's a + 2(1/2(b + c)), = a + b + c! :smile:
 

1. What is the Vector Proof Centroid Theorem?

The Vector Proof Centroid Theorem is a mathematical theorem that states that the centroid of a triangle is located at the intersection of its medians, which are the line segments connecting each vertex to the midpoint of the opposite side.

2. How is the Vector Proof Centroid Theorem proved?

The Vector Proof Centroid Theorem is typically proved using vector algebra and geometry. It involves showing that the vector sum of the medians is equal to zero, which indicates that they intersect at a single point.

3. What is the significance of the Vector Proof Centroid Theorem?

The Vector Proof Centroid Theorem is important in both mathematics and engineering, as it provides a way to determine the center of mass of a triangle. This helps in analyzing the stability and balance of structures and objects.

4. Can the Vector Proof Centroid Theorem be generalized to other shapes?

Yes, the Vector Proof Centroid Theorem can be extended to any polygon, not just triangles. It states that the centroid of a polygon is located at the intersection of its medians, which are the line segments connecting each vertex to the midpoint of the opposite side.

5. Are there any real-world applications of the Vector Proof Centroid Theorem?

Yes, the Vector Proof Centroid Theorem has many practical applications in fields such as engineering, architecture, and physics. It is used to determine the center of mass of objects and structures, which is important in designing stable and balanced systems.

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