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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
11 months ago

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This is 2012 IMO Problem 1. No chance that I can solve it :P Svatejas Shivakumar · 11 months ago

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@Svatejas Shivakumar Upload the solution Lakshya Sinha · 11 months ago

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@Lakshya Sinha 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. Svatejas Shivakumar · 11 months ago

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@Xuming Liang, gave a nice hint but here is the official solution Lakshya Sinha · 11 months ago

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@Lakshya Sinha More solutions can be found here Svatejas Shivakumar · 11 months ago

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Hint: There are cyclic shapes in the diagram. Prove that \(J\) is the circumcenter of \(AST\) Xuming Liang · 11 months ago

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