August 9th, 2010, 04:24 AM  #1 
Newbie Joined: Jun 2010 Posts: 22 Thanks: 0  Hard prove !
Hello ! Subjected to a square which has two isosceles triangles. One of them is: a triangle whose base is one of the sides of the square with base angles of 15. The other is a triangle whose base is the side opposite the base of the triangle it first touches the apex of the triangle at the end of the first node. Proved that the second triangle equilateral! 
August 9th, 2010, 04:45 AM  #2 
Senior Member Joined: Apr 2010 Posts: 105 Thanks: 0  Re: Hard prove !
Can you picture it?

August 9th, 2010, 06:01 AM  #3 
Senior Member Joined: Apr 2010 Posts: 105 Thanks: 0  Re: Hard prove !
Try to find tan(15) with the doubleangle formulae. So, 1/?3=tan(30)=... What has this to do with your problem.? By the way, there are 4 isosceles triangles? 
August 9th, 2010, 03:21 PM  #4  
Global Moderator Joined: Dec 2006 Posts: 19,162 Thanks: 1638  Quote:
 
August 9th, 2010, 05:41 PM  #5 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,821 Thanks: 1047 Math Focus: Elementary mathematics and beyond  Re: Hard prove !
Trigonometry: Let the sides of the square be of length b units. Construct a segment from P perpendicular to AB meeting AB at point Q and passing through P to DC meeting DC at S. Then PQ = b * tan(15°)/2. tan(PDS) = ((2b  b * tan(15°))/2)/(b/2) = 2  tan(15°). By the tangent halfangle formula tan(15°) = csc(30°)  cot(30°) = 2  ?(3), so the tangent of angle PDS = ?(3), so angle PDS = 60°. By symmetry angle SCP is 60° so triangle PDC is equilateral. 
August 9th, 2010, 07:33 PM  #6  
Newbie Joined: Jun 2010 Posts: 22 Thanks: 0  Re: Hard prove ! Quote:
 
August 9th, 2010, 07:39 PM  #7 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,821 Thanks: 1047 Math Focus: Elementary mathematics and beyond  Re: Hard prove ! 
August 9th, 2010, 08:08 PM  #8  
Newbie Joined: Jun 2010 Posts: 22 Thanks: 0  Re: Hard prove ! Quote:
 
August 9th, 2010, 08:37 PM  #9  
Newbie Joined: Jun 2010 Posts: 22 Thanks: 0  Re: Quote:
You are the king of geometric ! Thanks a lot friend !!!  

Tags 
hard, prove 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
A really hard math problem "Prove that if..."  isel  Algebra  8  September 22nd, 2013 09:35 AM 
hard Trigonometry prove  stuart clark  Algebra  1  March 23rd, 2011 11:46 PM 
Goldbach's conjecture (to prove or not to prove)  octaveous  Number Theory  13  September 23rd, 2010 04:36 AM 
prove prove prove. currently dont know where to post  qweiop90  Algebra  1  July 31st, 2008 06:27 AM 
prove prove prove. currently dont know where to post  qweiop90  New Users  1  December 31st, 1969 04:00 PM 