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August 25th, 2010, 04:40 PM  #1 
Senior Member Joined: Aug 2010 Posts: 109 Thanks: 0  Find lengths of the triangle ? 
August 25th, 2010, 04:41 PM  #2 
Senior Member Joined: Aug 2010 Posts: 109 Thanks: 0  Re: Find lengths of the triangle ?
Find lengths of the triangle A B C ?

August 25th, 2010, 06:53 PM  #3 
Senior Member Joined: Aug 2008 From: Blacksburg VA USA Posts: 338 Thanks: 4 Math Focus: primes of course  Re: Find lengths of the triangle ?
we like to see demonstrated effort  what have you tried or how do you think you might start?

August 25th, 2010, 11:11 PM  #4 
Global Moderator Joined: Dec 2006 Posts: 17,719 Thanks: 1359 
Hint: find and then use a very simple formula involving the inradius of a triangle.

August 26th, 2010, 11:06 AM  #5 
Senior Member Joined: Jan 2009 Posts: 344 Thanks: 3  Re: Find lengths of the triangle ?
Let's call the biggest triangle 3, the big triangle 2, and the small triangle 1. Let's call the big circle inscribed in triangle 2 C1 and the small circle inscribed in triangle 1 C2. If you have a compass and pencil try to reconstruct the drawing. Google how to inscribe a circle in a triangle. Try to find a solution through the process of inscribing the two circles in the two triangles. Don't worry about the radii of C1 and C2. At least try. This will enrich you. Don't worry about the radii. After construction, ask yourself, what should I do to inscribe a circle with a R=4 in triangle 2 and inscribe another circle with R=3 in triangle 2? 
August 27th, 2010, 09:30 AM  #6 
Senior Member Joined: Aug 2010 Posts: 109 Thanks: 0  Re: Find lengths of the triangle ?
hi What is the solution ? 
August 27th, 2010, 10:38 AM  #7 
Global Moderator Joined: Dec 2006 Posts: 17,719 Thanks: 1359 
Make a wild guess. It might be a lot closer than you'd expect to the correct answer.

August 27th, 2010, 10:57 AM  #8  
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 462 Math Focus: Calculus/ODEs  Re: Quote:
I found two ways to find this simple formula using similarity, and one way involving the halfangle identity for the tangent function. Make a drawing and see what you can come up with. Also, a right triangle divided into two smaller right triangles as in the diagram, will have a very famous relationship between the radii of the two smaller incircles and the radius of the incircle for the original right triangle, as well as all other corresponding parts.  
August 27th, 2010, 01:16 PM  #9 
Global Moderator Joined: Dec 2006 Posts: 17,719 Thanks: 1359 
Any method of calculating the inradius will suffice, but I had in mind a formula that gives a slight hint (not that it's really needed) as to what to do next. There are at least two formulas that do that, but one, though closely related to the other, is simpler and better known. Even a poor attempt at a more accurate diagram helps as well.

August 27th, 2010, 02:18 PM  #10 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 462 Math Focus: Calculus/ODEs  Re: Find lengths of the triangle ?
Ok, NOW I think I have found the formula you are referring to, as it does give a good indicator of how to proceed next. You are right, you can get the solution to this problem either way, but this other formula does point the way more clearly. Also, this better formula is more general as well, since when I set the two equal to each other, it reduces to the theorem of Pythagoras, confirming that the one I was using is specific to right triangles. As usual, I have learned from reading your posts. 

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