Determine-the-set-of-points-M-such-as-MA-MB-2MC-6-3-AB-BC-AC-6-ABC-is-triangle-

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

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] \dots .

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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