Here are a number of problems based around the same configuration:

The solutions I have posted make use of areal/barycentric co-ordinates (two names for the same thing). These are useful in problems involving ratios of lengths, areas and cevians (lines from the verticies of a triangle to the sides that are concurrent at a point).**Full Description**

Let \(O\) be a point in acute-angle triangle \(ABC\).

\(D\) is the intersection of \(AO\) and \(BC\). \(E,F\) are defined similarly.

\(X\) is the intersection of \(EF\) and \(AD\). \(Y,Z\) are defined similarly.

Let \(P\) be the intersection of \(XY\) and \(CF\) and \(Q\) be the intersection of \(XZ\) and \(BE\).

\(R\) is the intersection of \(AP\) with \(BC\) and \(S\) is the intersection of \(AQ\) with \(BC\).

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## Comments

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TopNewestThis looks nice!

Could you explain the configuration in detail, and the results you've obtained?

To me, it looks like the succesive medial triangles of \( \triangle ABC \), with common centroid \( G \).

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I've added a full description of the configuration. I've started to write the results I've derived as problems. Here are links to the first two:

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