Exercise_(−) ABC is a triangle. AB=AC=2 and BC=2(√2). I is midle of [BC]. J is a point such as AJ^(→) =(2/3)AI^(→) . J is the center of gravity of the triangle. 1)a) we define the set(T) of ∀ point M of plane: AM^2 +BM^2 +CM^2 =8. Show that BM^2 +CM^2 =2IM^2 +4 and AM^2 +2IM^2 =3JM^2 +(4/3) b)deduct that AM^2 +BM^2 +CM^2 =3JM^2 +((16)/3) c)Deduct the nature of (T)


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Question Number 97174 by mathocean1 last updated on 06/Jun/20

$\underline{E x e r c i s e}$ $A B C i s a t r i a n g l e . A B = A C = 2 a n d$ $B C = 2 \sqrt{2} . I i s m i d l e o f [B C] .$ $J i s a p o i n t s u c h a s \vec{A J} = \frac{2}{3} \vec{A I} . J i s$ $t h e c e n t e r o f g r a v i t y o f t h e t r i a n g l e .$ $1) a) w e d e f i n e t h e s e t (T) o f \forall p o i n t M$ $o f p l a n e :$ ${A M}^{2} + {B M}^{2} + {C M}^{2} = 8 .$ $S h o w t h a t {B M}^{2} + {C M}^{2} = 2 {I M}^{2} + 4$ $a n d {A M}^{2} + 2 {I M}^{2} = 3 {J M}^{2} + \frac{4}{3}$ $b) d e d u c t t h a t$ ${A M}^{2} + {B M}^{2} + {C M}^{2} = 3 {J M}^{2} + \frac{16}{3}$ $c) D e d u c t t h e n a t u r e o f (T)$
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