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### Medians of a Triangle

```
Date: 02/16/99 at 22:06:24
From: Jin Park
Subject: Medians of a Triangle

I learned in school that the 3 medians of a triangle divide themselves
up into a ratio of 1:2. I clearly understand what this means but could
you give me a proof of it (that the medians of triangle divide
themselves up into a ratio of 1:2)?  I guess I'm not stuck on anything
right now because I do not know where to start!

Thanks,
Jin Park
```

```
Date: 02/17/99 at 02:54:03
From: Doctor Floor
Subject: Re: Medians of a Triangle

Let us consider a triangle ABC. The midpoints of the sides form a
triangle A'B'C'. AA', BB' and CC' meet in a point G (known as the
centroid):

First we notice that triangle A'B'C' is similar to ABC; the sides of
A'B'C' have half the length of the the sidelengths of ABC. We can see
that, because for instance B'C' is an image of BC from multiplication
over A with factor 1/2. So indeed B'C' must be half the length of BC.

Also we notice that AA' passes through the midpoint of B'C', since in
the multiplication mentioned above, A' moves to the midpoint of B'C'.
Similar results are found for BB' and CC'. But that means that the
medians of ABC are also medians of A'B'C'.

We now can conclude that the distance AG in ABC corresponds to A'G in
A'B'C'. From the observation that A'B'C' has lengths half of those in
ABC, we can conclude that AG:A'G = 2:1. And that is what we wanted to
prove.

I hope this helps!

Best regards,
- Doctor Floor, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Geometry
High School Triangles and Other Polygons
Middle School Geometry
Middle School Triangles and Other Polygons

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