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February 2nd, 2019, 06:49 AM   #1
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Question 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
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February 2nd, 2019, 10:28 AM   #2
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Have you tried to define the various mid-points, centroids and intersections mentioned in the question in terms of the position vectors for A, B, C and D?
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February 2nd, 2019, 10:58 AM   #3
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Quote:
Originally Posted by skipjack View Post
Have you tried to define the various mid-points, centroids and intersections mentioned in the question in terms of the position vectors for A, B, C and D?
I don't understand how to use vectors and when it's okay. Can you be more specific? Thank you in advance!

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Kaloyan
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February 2nd, 2019, 11:44 AM   #4
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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.
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February 2nd, 2019, 12:06 PM   #5
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Quote:
Originally Posted by skipjack View Post
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.
I've just looked about position vectors, but I don't know how I can use them in the current problem. I don't understand vectors as I said because I think that there's always a different way of showing something. Can you show me the first step of the solution? I will be very grateful. Thank you in advance!

Greetings,
Kaloyan
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February 2nd, 2019, 03:09 PM   #6
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You didn't say what web site(s) you visited, but try this one, that tells you how to find position vectors for the mid-point of a line and the centroid of a triangle, making sure you fully understand how the results are obtained.
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