Question 805599: A triangle ABC has its 3 medians drawn in. There are three line segments formed starting at the centroid and going out (one to vertex C, one to midpoint of AC, and one to the midpoint of BC). The midpoints of these three segments are then connected to form a triangle. The area of this triangle is 2. What is the area of triangle ABC?
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website!  The area of the blue triangle is 2.
I will connect other midpoints and make a red, medium sized triangle.
 Now I have 3 similar triangles: 2 red triangles, and one blue triangle.
The sides of the red, medium sized triangle XYZ are twice as long as those of the small blue triangle.
The sides of triangle ABC are twice as long as those of the red, medium sized triangle XYZ,
and hence 4 times as long as those of the small blue triangle.
Do you want proof? See the note below.
A similar figure with all sides scaled up by a factor of 
has  times the area.
So large triangle ABC has sides times as long as the small blue triangle,
and an area times larger than the small blue triangle.
The area of triangle ABC is  .
WHY ONE TRIANGLE HAS SIDES TWICE AS LONG AS ANOTHER TRIANGLE:
Triangle ABW is similar to triangle XYW
because they have the same angle at W,
side XW is half as long as side AW,
and side YW is half as long as side BW.
Because they are similar triangles,
the third sides are in the same ratio,
so side XY is half as long as side AB.
That is the story for triangles ABW and XYW.
We can tell a similar story for triangles ACW and XZW,
and we can tell a similar story for triangles BCW and YZW.
Adding the three stories,we get that
side XY is half as long as side AB,
side XZ is half as long as side AC, and
side YZ is half as long as side BC.
That is the end of the story for triangles ABC and XYZ.
The sides of triangle XYZ are half as long as those of triangle ABC,
so the sides of triangle ABC are twice as long as those of XYZ, as I had said above.
You can add a similar story comparing the blue triangle to triangle XYZ.
You would have to add a couple more letters to name the two unnamed vertices of the blue triangle,
but you can make a good alphabet soup with all those letters when you are done.
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