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

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)