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Found problems: 1

2012 Bogdan Stan, 4
Tags: vecotrial geometry , geometry
Let be three real positive numbers $ \alpha ,\beta ,\gamma $ and let $ M,N $ be points on the sides $ AB,BC, $ respectively, of a triangle $ ABC, $ such that $ \frac{MA}{MB} =\frac{\alpha }{\beta } $ and $ \frac{NB}{NC} =\frac{\beta }{\gamma } . $ Also, let $ P $ be the intersection of $ CM $ with $ AN. $ Show that: $$ \frac{1}{\alpha }\overrightarrow{PA} +\frac{1}{\beta }\overrightarrow{PB} +\frac{1}{\gamma }\overrightarrow{PC} =0 $$