Given:
In #DeltaABC,# #D# is the mid point of #BC# i.e.#AD# is the median.
RTP:#" "" "" "" "" "AB^2+AC^2=AD^2+BD^2#
Construction:#" " " "AE_|_BC# is drawn at #E#
PROOF:#" "DeltaABE# is right angled at #E# ,
So by Pythagoras theorem we have
#AB^2=AE^2+BE^2.......[1]#
Similarly for #DeltaADE# we have
#AC^2=AE^2+CE^2.......[2]#
And for #DeltaACE# we have
#AD^2=AE^2+DE^2.......[3]#
Adidng [1] and [2] we get
#AB^2+AC^2=2AE^2+BE^2+CE^2#
#=>AB^2+AC^2=2AE^2+(BD-DE)^2+(CD+DE)^2#
#=>AB^2+AC^2=2AE^2+BD^2+DE^2-2BD*DE+CD^2+DE^2+2CD*DE#
#=>AB^2+AC^2=2AE^2+BD^2+2DE^2-cancel(2BD*DE)+BD^2+cancel(2BD*DE)#
Here we used the fact that #D# is halfway between #B# and #C# so #2color(blue)(BD)*DE=2color(blue)(CD)*DE#.
#=>AB^2+AC^2=2AE^2+2BD^2+2DE^2#
#=>AB^2+AC^2=2(AE^2+DE^2)+2BD^2#
#=>color(red)(AB^2+AC^2=2AD^2+2BD^2)#
This is known as Apollonius's theorem
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