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October 29th, 2013, 12:50 PM  #1 
Newbie Joined: Sep 2013 Posts: 10 Thanks: 0  Pythagoras problem
I tried using Apollonius's theorem to relate v, w and CP. Let the side opposite <B be b and <A be a and v+w be c. Let D be the point on AB where the altitude from <C falls and PD=y. By Pythagoras: b^2+a^2=v^2+w^2+2CP^2 =(vy)^2+(v+y)^2+2CP^2 =2(v+y)^2+2y^2+2CP^2 this is where I get stuck I see that b^2+a^2=(v+w)^2 but I can't simplify it to get that v^2+w^2=2CP^2 
October 29th, 2013, 06:19 PM  #2 
Senior Member Joined: Jul 2011 Posts: 118 Thanks: 0  Re: Pythagoras problem 
October 29th, 2013, 06:42 PM  #3 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,968 Thanks: 1152 Math Focus: Elementary mathematics and beyond  Re: Pythagoras problem From the law of cosines, Similarly, Adding these results, 
October 29th, 2013, 09:01 PM  #4 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,968 Thanks: 1152 Math Focus: Elementary mathematics and beyond  Re: Pythagoras problem
Construct a segment from C to AB, bisecting AB at D. CP² = (AD  v)² + (w  PD)² = AD²  2vAD + v² + w²  2wPD + PD² = v² + w² + CP²  2(vAD + wPD) = v² + w² + CP²  2(v² + 2vPD + 2PD²) [as AD = PD + v and w = 2PD + v] = v² + w² + CP²  2((v + PD)² + PD²) = v² + w² + CP²  2CP² 2CP² = v² + w². 
October 29th, 2013, 09:15 PM  #5 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1039  Re: Pythagoras problem
Skinning the cat #3! Code: C A P D B 2a^2 = v^2 + 2vw + w^2 [1] v + y = w  y : y = (w  v) / 2 [2] d^2 = p^2  y^2 [3] d^2 = a^2  (w  y)^2 : d^2 = a^2  w^2 + 2wy  y^2 [4] [3][4]: p^2 = a^2  w^2 + 2wy Substitute [2]: p^2 = a^2  w^2 + 2w((w  v)/2) : 2p^2 = 2a^2  2vw Substitute [1]: 2p^2 = v^2 + 2vw + w^2  2vw : 2p^2 = v^2 + w^2 
October 30th, 2013, 02:13 AM  #6 
Global Moderator Joined: Dec 2006 Posts: 21,026 Thanks: 2257 
Construct PD perpendicular to BC with D on BC. It is easy to show that PD = w/?2 and CD = v/?2. By Pythagoras, v²/2 + w²/2 = CP², so v² + w² = 2CP². 

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