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February 2nd, 2019, 06:49 AM  #1 
Newbie Joined: Feb 2019 From: Bulgaria Posts: 3 Thanks: 0  The National Mathematical Olympiad in Bulgaria
We have the quadrilateral ABCD. The middles of AB, BC, CD and DA, are respectively M, N, P and Q. The centroid of BNP is F, and the centroid of NPD is G. MG intersects FQ at K. We know that FK = 6 and we have to show KQ = 9. Thank you in advance and sorry for my bad English. I made drawing: https://prnt.sc/mfox41 
February 2nd, 2019, 10:28 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,469 Thanks: 2038 
Have you tried to define the various midpoints, centroids and intersections mentioned in the question in terms of the position vectors for A, B, C and D?

February 2nd, 2019, 10:58 AM  #3  
Newbie Joined: Feb 2019 From: Bulgaria Posts: 3 Thanks: 0  Quote:
Greetings, Kaloyan  
February 2nd, 2019, 11:44 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 20,469 Thanks: 2038 
A vector can be thought of as a line with particular direction and length, but not necessarily a fixed position. The vectors from some fixed point (not necessarily named) to various points of interest are called the position vectors of those points. If you do a Google search for "using position vectors", you should find a lot of articles and videos that explain the basics adequately for this problem. 
February 2nd, 2019, 12:06 PM  #5  
Newbie Joined: Feb 2019 From: Bulgaria Posts: 3 Thanks: 0  Quote:
Greetings, Kaloyan  
February 2nd, 2019, 03:09 PM  #6 
Global Moderator Joined: Dec 2006 Posts: 20,469 Thanks: 2038 
You didn't say what web site(s) you visited, but try this one, that tells you how to find position vectors for the midpoint of a line and the centroid of a triangle, making sure you fully understand how the results are obtained.


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