In the triangle ABC we draw the median segment AD and the bisector BE. We know that AB=7, BC=18 ed EA=ED. How is the length of AC?
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Using Geogebra I obtained $15$, but I don't know yet how to obtain it with a proper reasing. – ajotatxe Jun 18 '17 at 10:28
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The bisector BE, is that a perpendicular bisector or an angle bisector? – Lundborg Jun 18 '17 at 10:44
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It is the bisector of the angle B. – G. Gadda Jun 18 '17 at 10:48
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is something else given in the triangle? – Dr. Sonnhard Graubner Jun 18 '17 at 10:53
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No, that is all. – G. Gadda Jun 18 '17 at 10:54
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Since $\measuredangle ABE=\measuredangle DBE$ and $AE=DE$,
we see that $BAED$ is cyclic, which says that $\Delta ABC\sim\Delta DEC$.
Let $AE=7x$.
Hence, $ED=7x$ and $EC=18x$ and since $$\frac{ED}{AB}=\frac{DC}{AC},$$ we obtain $$\frac{7x}{7}=\frac{9}{25x},$$ which gives $x=\frac{3}{5}$ and $AC=25\cdot\frac{3}{5}=15$.
Done!
Michael Rozenberg
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Thank you very much! But I do not understand the equivalence between segments that leads to the solving equation. – G. Gadda Jun 18 '17 at 11:20
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@G. Gadda It's just substitution of $ED=7x$, $DC=9$, $AB=7$ and $AC=25x$ to $\frac{ED}{AB}=\frac{DC}{AC}$. – Michael Rozenberg Jun 18 '17 at 11:30
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Why the ratio between DC and AC is equal to the ratio between ED and AB? Thank you. – G. Gadda Jun 18 '17 at 11:33
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For this reason I thought to put in the right part of the equation the ratio between DC and BC... but I know you are right checking the result, just I do not unerstand why.. sorry, I am little confused :) – G. Gadda Jun 18 '17 at 11:38
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