Prove that 3 times the sum of squares of sides of a triangle is equal to four times the sum of squares of its medians.

Asked by pkirankumar4321 | 19th Sep, 2015, 07:18: AM

Expert Answer:

 
 
begin mathsize 14px style T o space p r o v e colon 3 left parenthesis A B squared plus B C squared plus A C squared right parenthesis equals 4 left parenthesis A D squared plus B E squared plus C F squared right parenthesis  P r o o f colon I n space t r i a n g l e space B A C comma space A D comma space B E space a n d space C F space a r e space m e d i a n s. B y space A p p o l o n i u s space t h e o r e m comma space w e space h a v e A B squared plus A C squared equals 2 A D squared plus 2 B D squared space space space space space space space space... left parenthesis 1 right parenthesis A B squared plus B C squared equals 2 B E squared plus 2 E C squared space space space space space space space space space space.... left parenthesis 2 right parenthesis A C squared plus B C squared equals 2 F C squared plus 2 A F squared space space space space space space space space.... left parenthesis 3 right parenthesis A d d i n g space left parenthesis 1 right parenthesis comma space left parenthesis 2 right parenthesis space a n d space left parenthesis 3 right parenthesis comma space w e space g e t A B squared plus A C squared plus A B squared plus B C squared plus A C squared plus B C squared equals 2 A D squared plus 2 B D squared plus 2 B E squared plus 2 E C squared plus 2 F C squared plus 2 A F squared 2 left parenthesis A B squared plus B C squared plus A C squared right parenthesis equals 2 left parenthesis A D squared plus B E squared plus F C squared right parenthesis plus 2 left parenthesis A F squared plus B D squared plus E C squared right parenthesis 2 left parenthesis A B squared plus B C squared plus A C squared right parenthesis equals 2 left parenthesis A D squared plus B E squared plus F C squared right parenthesis plus 2 left parenthesis open parentheses fraction numerator A B over denominator 2 end fraction close parentheses squared plus open parentheses fraction numerator B C over denominator 2 end fraction close parentheses squared plus open parentheses fraction numerator A C over denominator 2 end fraction close parentheses squared right parenthesis 2 left parenthesis A B squared plus B C squared plus A C squared right parenthesis equals 2 left parenthesis A D squared plus B E squared plus F C squared right parenthesis plus 2 over 4 left parenthesis A B squared plus B C squared plus A C squared right parenthesis 2 left parenthesis A B squared plus B C squared plus A C squared right parenthesis equals 2 left parenthesis A D squared plus B E squared plus F C squared right parenthesis plus 1 half left parenthesis A B squared plus B C squared plus A C squared right parenthesis  left parenthesis A B squared plus A C squared plus B C squared right parenthesis open parentheses 3 over 2 close parentheses equals 2 left parenthesis A D squared plus B E squared plus C F squared right parenthesis 3 left parenthesis A B squared plus A C squared plus B C squared right parenthesis equals 4 left parenthesis A D squared plus B E squared plus C F squared right parenthesis   end style

Answered by Rashmi Khot | 20th Sep, 2015, 03:47: PM

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