For discussing Olympiad level Geometry Problems
nafistiham
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let $O$ be the centre of the circle through the points $A,B,C$.Let $D$be the mid point of $AB$.Let $E$ be the centroid of the triangle $ACD$.Prove that, the line $CD$ will be perpendicular to the line $OE$ iff $AB=AC$

the book provides the solution in vector could someone do it in the geometric way?

$7777^{th}$ Last edited by nafistiham on Mon Jan 23, 2012 6:23 pm, edited 1 time in total.
$\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0$
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Avik Roy
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### Re: Second Balkan Mathematical Olympiad

প্রশ্নই তো অসম্পূর্ণ মনে হচ্ছে!!!
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nafistiham
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### Re: Second Balkan Mathematical Olympiad

nafistiham wrote:Prove that, the line $CD$ will be perpendicular to the line $OE$ iff $AB+AC$
sorry. it should have been like this.
nafistiham wrote:Prove that, the line $CD$ will be perpendicular to the line $OE$ iff $AB=AC$
$\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0$
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please co-operate.

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### Re: Second Balkan Mathematical Olympiad Geo 3.pdf
It's very easy to prove $OE\perp CD$ when $AB=AC$.So I'm giving its inverse case.
My Proof :
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nafistiham
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### Re: Second Balkan Mathematical Olympiad

thanks for the nice proof. (and i am uploading the figure as a jpeg file, so that none has to download it. ) from a previous post by phlembac adib hasan
Geo 3.JPG (40.55 KiB) Viewed 1835 times
$400^{th}$
$\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0$
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please co-operate.