Dominion Strategy Forum

Miscellaneous => General Discussion => Topic started by: heron on February 13, 2013, 08:37:10 pm

Title: Geometry Problem
Post by: heron on February 13, 2013, 08:37:10 pm

The medians AD, BE, and CF of triangle ABC intersect at the centroid G. The line through G that is parallel to BC intersects AB and AC at M and N, respectively. If the area of triangle ABC is 144, then find the area of triangle ENG.

So, I received this problem on Art of Problem Solving's Alcumus program, and I don't really have a clue as to how to solve it. I drew the picture, and I don't really see why there aren't multiple solutions. I struggle with any problems with words such as median, centroid, altitude, incenter, etc.

So, any ideas here?

Title: Re: Geometry Problem
Post by: DG on February 13, 2013, 09:09:33 pm

• Each median divides the triangle into two smaller triangles of equal area.
• The centroid is exactly two-thirds the way along each median

These two googled facts about the centroid let you work out the ratios of areas of triangles in the question (without knowing the length of any side).

Title: Re: Geometry Problem
Post by: heron on February 13, 2013, 09:20:23 pm

Well, it seems that works, the answer was 8. But it doesn't feel right. I keep think that if I have a long, skinny triangle I'll be able to evade the ratios.
I guess the googled fact that I was missing was "The three medians' long sides divide the triangle into 3 equal triangles. Although I suppose that can be determined by the other two facts.

Thanks!