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Proof Contest Day 4

In the triangle \(ABC\) the point \( J\) is the center of the excircle opposite to \( A\). This excircle is tangent to the side \(BC\) at \( M\), and to the lines \( AB\) and \(AC\) at \(K\) and \(L\) respectively. The lines \(LM \) and \(BJ\) meet at \(F\), and the lines \(KM \) and \(CJ \) meet at \(G\). Let \(S\) be the point of intersection of the lines \(AF\) and \( BC\), and let \(T\) be the point of intersection of the lines \(AG\) and \(BC\). Prove that \(M\) is the midpoint of \(ST\).


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Note by Lakshya Sinha
1 year, 10 months ago

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This is 2012 IMO Problem 1. No chance that I can solve it :P

Brilliant Member - 1 year, 10 months ago

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Upload the solution

Lakshya Sinha - 1 year, 10 months ago

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I did not get what you mean. I have not solved the problem. Just copy and paste this problem in google and you will find the solutions when you click the Aops if you were asking me for that.

Brilliant Member - 1 year, 10 months ago

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@Xuming Liang, gave a nice hint but here is the official solution

Lakshya Sinha - 1 year, 10 months ago

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More solutions can be found here

Brilliant Member - 1 year, 10 months ago

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Hint: There are cyclic shapes in the diagram. Prove that \(J\) is the circumcenter of \(AST\)

Xuming Liang - 1 year, 10 months ago

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