http://jwilson.coe.uga.edu/EMAT6680Fa11/Antillanca/EMAT6680.gif


Rayen Antillanca. Assignment 4


 

The centroid of the triangle is the intersection of the three medians. Recall, a median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. Therefore a triangle has three medians.

http://jwilson.coe.uga.edu/EMAT6680Fa11/Antillanca/Assignment4/centroid.gif

 

 


 

The next explanation proves that these three medians are concurrent.

 

Step 1

Let ABC be any triangle, and let D, E, and F be midpoints for the sides AC, AB and BC, respectively. The medians from B and C meet at point G.

dem1.gif

 

 

 

Step 2

Now, I draw a line parallel to DC through F, and call Q its intersection with side AB

dem2.gif

 

 

Step 3

The two triangles, ∆AQF and ∆ADC, are similar, and so AF:AC:AQ:AD, which implies that Q is the midpoint of AD.

But AD=DB, so that QD:DB::1:2 and since ∆QBF and ∆DBG are similar, the point G lies on median FB at one-third its length from F.

dem3.gif

 

Step 4

Let K be the intersection of the two medians AE and BF. I draw a line parallel to AE through F, and call R its intersection with BC. The same argument in step 3 is used here to conclude that the point K lies on median FB, at one-third its length from point F

dem4.gif

 

Step 5

This means that the two points, H and K, are one and the same, so that the three medians are concurrent at a point, say G. This completes the proof. G is the centroid.

dem5.gif

 

 

Summary

The Centroid (G) divides each median in the ratio 2:1, then the segment from a vertex to the centroid is 2/3 of the median from that vertex.

To see an example in GSP of this fact clic here

 

 

Note: All figures on this web page were made with The Geometerís Sketchpad

 

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