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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 $$ST$$.

Not original

Note by Lakshya Sinha
2 years, 1 month ago

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

- 2 years, 1 month ago

- 2 years, 1 month ago

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.

- 2 years, 1 month ago

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

- 2 years, 1 month ago

More solutions can be found here

- 2 years, 1 month ago

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

- 2 years, 1 month ago