Proof Contest Day 4

In the triangle ABCABC the point J J is the center of the excircle opposite to A A. This excircle is tangent to the side BCBC at M M, and to the lines AB AB and ACAC at KK and LL respectively. The lines LMLM and BJBJ meet at FF, and the lines KMKM and CJCJ meet at GG. Let SS be the point of intersection of the lines AFAF and BC BC, and let TT be the point of intersection of the lines AGAG and BCBC. Prove that MM is the midpoint of STST.


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Note by Department 8
3 years, 9 months ago

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

A Former Brilliant Member - 3 years, 9 months ago

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Upload the solution

Department 8 - 3 years, 9 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.

A Former Brilliant Member - 3 years, 9 months ago

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Hint: There are cyclic shapes in the diagram. Prove that JJ is the circumcenter of ASTAST

Xuming Liang - 3 years, 9 months ago

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

Department 8 - 3 years, 9 months ago

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More solutions can be found here

A Former Brilliant Member - 3 years, 9 months ago

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