Determine the set of points M such as ∣∣MA^→ +MB^→ +2MC^→ ∣∣=6(√3) AB=BC=AC=6 ABC is triangle.


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Question Number 79131 by mathocean1 last updated on 23/Jan/20

$Determine the set of points M$ $such as ∣∣ M \vec{A} + M \vec{B} + 2 M \vec{C} ∣∣= 6 \sqrt{3}$ $AB = BC = AC = 6$ $ABC is triangle .$
Commented by mathocean1 last updated on 23/Jan/20

$please help me sirs$
Commented by msup trace by abdo last updated on 23/Jan/20
$let G =barycentre of the system {(A,1),(B,1) ,(C,2)} (e) ⇔∣∣4MG^→ ∣∣=6(√3)⇒MG=((6(√3))/4) ⇒MG=((3(√3))/2) ⇒M varie on cercle C(G,((3(√3))/2)) how to trace G? G is bsrycentre of { (I,2),(C,2)} with I midpoint of [A,B] ⇒G midpoint of [I,C]....$
$l e t G = b a r y c e n t r e o f t h e s y s t e m$ ${(A, 1), (B, 1), (C, 2)}$ $(e) \Leftrightarrow∣∣ 4 M \vec{G} ∣∣= 6 \sqrt{3} \Rightarrow M G = \frac{6 \sqrt{3}}{4}$ $\Rightarrow M G = \frac{3 \sqrt{3}}{2} \Rightarrow M v a r i e o n c e r c l e$ $C (G, \frac{3 \sqrt{3}}{2}) h o w t o t r a c e G ?$ $G i s b s r y c e n t r e o f {(I, 2), (C, 2)}$ $w i t h I m i d p o i n t o f [A, B] \Rightarrow G$ $m i d p o i n t o f [I, C] . . . .$
	Answered by ~blr237~ last updated on 23/Jan/20

	$l e t I b e t h e m i d d l e o f [A B] a n d J t h e m i d d l e o f [I C]$ $l e t r e d u c e \vec{u} = M \vec{A} + M \vec{B} + 2 M \vec{C}$ $\vec{u} = M \vec{I} + I \vec{A} + M \vec{I} + I \vec{B} + 2 M \vec{C}$ $= 2 M \vec{I} + 2 M \vec{C} c a u s e I \vec{B} = - I \vec{A}$ $= 2 (M \vec{J} + J \vec{I} + M \vec{J} + J \vec{C})$ $= 4 M \vec{J} c a u s e J m i d d l e o f [I C]$ $T h e s e a r c h p o i n t m u s t s a t i s f y ∣∣ 4 M \vec{J} ∣∣= 6 \sqrt{3} \Leftrightarrow M J = \frac{3 \sqrt{3}}{2}$ $F i n a l y t h e s e a r c h e d s e t i s t h e c i r c l e w h o s e c e n t e r i s J a n d r a d i u s i s \frac{3 \sqrt{3}}{2}$
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