August 26th, 2011, 05:02 PM  #1 
Senior Member Joined: Apr 2011 From: Recife, BR Posts: 352 Thanks: 0  Hard geometry
Let H, I, O be orthocenter/incenter/circumcenter of triangle ABC respectively. Let L be intersection of line CI and circumcircle such that L is not C. Given: AH = OH and IL = AB. Find all internal angles of ABC. This problem has kept my attention for at least two hours now.. I find it very hard to use the given information. All I have so far is: and Here denotes measurement of angle, the symbol for latex is not so good. Any hints are very appreciated 
August 30th, 2011, 05:26 AM  #2  
Senior Member Joined: Jun 2011 Posts: 298 Thanks: 0  Re: Hard geometry Quote:
 
August 30th, 2011, 08:56 AM  #3  
Senior Member Joined: Apr 2011 From: Recife, BR Posts: 352 Thanks: 0  Re: Hard geometry Quote:
I'll soon upload picture.  
August 30th, 2011, 08:26 PM  #4 
Senior Member Joined: Jun 2011 Posts: 298 Thanks: 0  Re: Hard geometry
Toughy!

August 31st, 2011, 09:01 AM  #5  
Senior Member Joined: Apr 2011 From: Recife, BR Posts: 352 Thanks: 0  Re: Hard geometry Quote:
1) AL = IL = BL; 2) Notice that AOBH is cyclic (which was the hardest part to me, I never see these things). Then it is trivial, I will not post answer so you can try it Here goes another one, just as hard and just as nice. I couldn't do this yet. is a triangle with and . The incircle touches and at and respectively. Let be the circumference with diameter . The line intersects at points: and . Find the distance between the midpoint of and the midpoint of . Geometry problems like these remind me of P = NP.. it is very hard to find solution, but very easy to check them  
September 1st, 2011, 05:42 AM  #6 
Senior Member Joined: Jun 2011 Posts: 298 Thanks: 0  Re: Hard geometry
Thanks. I am trying it by using vectors. It seems a bit cleaner. Example: Let AC intersect BH at P Then look for one more equation related to and Goodness! Math by nature is hard. I mean it can get very, very messy. 

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geometry, hard 
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