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June 8th, 2010, 12:44 AM  #1 
Newbie Joined: Nov 2009 Posts: 12 Thanks: 0  geometrical question about congruent rectangles
Hello, I have a geometrical question: there are two congruent rectangles, now you lay one on top of the other, so that they have eight shared points. It has to be shown that the area of the intersection is bigger than the area of half of the rectangle. Thanks marten. 
June 8th, 2010, 02:26 AM  #2 
Senior Member Joined: Apr 2010 Posts: 105 Thanks: 0  Re: geometrical question about congruent rectangles
How can I understand eight shared points, if there is an intersection of the areas?

June 8th, 2010, 06:14 AM  #3 
Newbie Joined: Nov 2009 Posts: 12 Thanks: 0  Re: geometrical question about congruent rectangles
Yes, there is an intersection of areas, I tried to upload an image, but it didn't work. I don't know if shared is the right word, perhaps "in common" is better ?!? 
June 8th, 2010, 06:21 AM  #4  
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: geometrical question about congruent rectangles Quote:
"Consider a rectangle in a plane. Rotate the rectangle slightly (let's say 2030 degrees) from its center so that the edges of the image intersect the edges of the preimage in 8 points. Show that (something about areas)..."  
June 8th, 2010, 06:28 AM  #5 
Senior Member Joined: Apr 2010 Posts: 105 Thanks: 0  Re: geometrical question about congruent rectangles
OK, the eight points are shared on the edges. Otherwise you have infinitely points too in the intersection of the areas.

June 8th, 2010, 08:44 AM  #6 
Global Moderator Joined: Dec 2006 Posts: 20,931 Thanks: 2207 
This is problem 12 from the 4th ASU (All Soviet Union Mathematical Olympiad) 1970. The original wording is as shown below. Two congruent rectangles of area A intersect in eight points. Show that the area of the intersection is more than A/2. 
June 8th, 2010, 08:57 AM  #7  
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Quote:
 
June 22nd, 2010, 12:04 AM  #8 
Newbie Joined: Nov 2009 Posts: 12 Thanks: 0  Re: geometrical question about congruent rectangles
no further ideas so far?

June 22nd, 2010, 01:57 AM  #9  
Senior Member Joined: Apr 2010 Posts: 105 Thanks: 0  Re: geometrical question about congruent rectangles Quote:
 
June 22nd, 2010, 04:26 AM  #10  
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: geometrical question about congruent rectangles Quote:
You have to choose two arbitrary rectangles that satisfy the conditions and then show something based on assumptions. Starting with squares leads to a weaker statement.  

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