Three lines, parallel to the sides of a triangle intersect in one point, and the segments of these three lines that are inside the triangle all have lengths equal to x. Evaluate x if the sides of the triangle are a,b,c. I've tried some stuff like similarity. How do I solve this?
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"I've tried some stuff like similarity"...and? – marco trevi Sep 26 '14 at 14:43
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Not much. I'm not sure how to relate a,b,c to x. – Jackie Poehler Sep 26 '14 at 14:47
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Are you saying that the segments of the lines inside the triangle all have length equal to $x$? I think that ordinarily, a line or segment can’t be equal to a number. – Lubin Sep 26 '14 at 14:55
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yeah, exactly. i'll change it – Jackie Poehler Sep 26 '14 at 14:57
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Assuming that the three segments intersect in $P$, we have, by similarity: $$ d(P,BC) = h_A\left(1-\frac{x}{a}\right)\tag{1}$$ where $h_A=d(A,BC)$ and $a=BC$. Since: $$ 2\Delta = \sum_{cyc} a\cdot d(P,BC)\tag{2} $$ it follows that: $$ 2\Delta = \sum_{cyc}ah_a - x\sum_{cyc}h_a\tag{3}$$ so: $$ x = \frac{4\Delta}{\sum_{cyc}h_a} = \frac{4\Delta}{\sum_{cyc}\frac{2\Delta}{a}}=\color{red}{\frac{2}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}}.$$
Jack D'Aurizio
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Now I get that delta means the area of a triangle. But why (1)? – Jackie Poehler Sep 26 '14 at 16:48
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