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October 29th, 2013, 12:50 PM   #1
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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
=(v-y)^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
Attached Images image.jpg (49.7 KB, 169 views) 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. CP� = (AD - v)� + (w - PD)� = AD� - 2vAD + v� + w� - 2wPD + PD� = v� + w� + CP� - 2(vAD + wPD) = v� + w� + CP� - 2(v� + 2vPD + 2PD�) [as AD = PD + v and w = 2PD + v] = v� + w� + CP� - 2((v + PD)� + PD�) = v� + w� + CP� - 2CP� 2CP� = v� + w�. 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 AC = BC = a (hate "s"!), AP = v, PD = y, BD = w - y, CP = p, CD = d 2a^2 = v^2 + 2vw + w^2  v + y = w - y : y = (w - v) / 2  d^2 = p^2 - y^2  d^2 = a^2 - (w - y)^2 : d^2 = a^2 - w^2 + 2wy - y^2  : p^2 = a^2 - w^2 + 2wy Substitute : p^2 = a^2 - w^2 + 2w((w - v)/2) : 2p^2 = 2a^2 - 2vw Substitute : 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, v�/2 + w�/2 = CP�, so v� + w� = 2CP�. Tags problem, pythagoras Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post stevembe Algebra 10 September 23rd, 2013 08:25 AM nadiva8685 Algebra 3 January 22nd, 2013 04:37 PM kaushiks.nitt Number Theory 2 June 27th, 2009 06:51 PM graviton120 Algebra 3 October 31st, 2008 08:04 PM jsmars Algebra 1 May 26th, 2008 01:53 AM

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