Constructing a Triangle from its medians and the relationship to the parent triangle.

Before any relationship could be determined, I explored the different possibilities of similarity, congruence, ratio of perimeters, and ratio of areas between an arbitrary triangle and a triangle constructed from its medians. Below are the GSP sketches and analytical data:

From the sketches, I was able to determine that there was no similarity or congruence relationships related to these two triangles. Upon further inspection, the only significant relationship determined was that the area of the triangle constructed from the medians of the parent triangle was smaller by a factor of 3/4 or 0.75. After further research into this discovery, this was a valid claim, and a provable relationship by using various methods.

Here is one proof:

In order to understand the proof, some background information needs to be presented.

Background:

Apollonius's theorem states that for sides, a,b,c in any triangle and m the median of side a let :

By rearranging the equation we can get:

2*m = [(2b^2) +(2c^2)-(a^2)]^(1/2)

By adding the other medians we can get the equation:

(m1^2)+(m2^2)+(m3^2) = (3/4)[(a^2)+(b^2)+(c^2)]

Here are a couple links that may be interesting with regards to some relationships found by other people:

• Angle relationships of median triangles and parent triangles. "Properties of Median Triangles," by Kelli Nipper.

http://jwilson.coe.uga.edu/EMT668/EMT668.Folders.F97/Nipper/Assignment%206/medianposition.html

• Median triangle, from Mathworld © 1996-9 Eric W. Weisstein
1999-05-26
: http://mathworld.pdox.net/math/m/m164.htm
• Proof by Samuel Obara, "The area of the median triangle is 75% of the original triangle ABC."

http://jwilson.coe.uga.edu/EMT668/EMAT6680.2000/Obara/Obaraassig6/proof%20tring%20med.html