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Question Number 80809 by mr W last updated on 06/Feb/20

Commented by mr W last updated on 06/Feb/20
$I is a point inside the triangle ABC. P,Q,R are the midpoint of AI,BI,CI respectively. What fraction of the triangle ABC does the shaded hexagon present?$
$I i s a p o i n t i n s i d e t h e t r i a n g l e A B C .$ $P, Q, R a r e t h e m i d p o i n t o f A I, B I, C I$ $r e s p e c t i v e l y .$ $W h a t f r a c t i o n o f t h e t r i a n g l e A B C$ $d o e s t h e s h a d e d h e x a g o n p r e s e n t ?$
Commented by ajfour last updated on 07/Feb/20

Commented by ajfour last updated on 07/Feb/20

$△_{A B C} = Σ 2 q r \sin α$ $e q . o f A R : \bar{v} = \bar{r} + λ (2 \bar{p} - \bar{r})$ $e q . o f C P : \bar{v} = \bar{p} + μ (2 \bar{r} - \bar{p})$ ${\bar{v}}_{E} = \bar{r} + λ (2 \bar{p} - \bar{r}) = \bar{p} + μ (2 \bar{r} - \bar{p})$ $\Rightarrow 1 - λ = 2 μ, 2 λ = 1 - μ$ $\Rightarrow 1 - λ = 2 - 4 λ \Rightarrow λ = 1 / 3$ ${\bar{v}}_{E} = \bar{r} + \frac{2 \bar{p} - \bar{r}}{3} = \frac{2}{3} (\bar{p} + \bar{r})$ $A_{I R E P} = p r \sin β (\frac{4}{9} + \frac{1}{9} + \frac{1}{9})$ $= \frac{2}{3} p r \sin β$ $A_{h e x a g o n} = \frac{2}{3} Σ p r \sin β$ $h e n c e \frac{A_{h e x a g o n}}{△_{A B C}} = \frac{2 / 3}{2} = \frac{1}{3} .$
Commented by mr W last updated on 07/Feb/20

$a n s w e r i s c o r r e c t!$ ${i t}^{'} s a n i c e v e c t o r m e t h o d s i r!$
Commented by ajfour last updated on 07/Feb/20

	Answered by mr W last updated on 07/Feb/20

	Commented by mr W last updated on 07/Feb/20

	$a n o t h e r w a y :$ $a s e x a m p l e w e t r e a t o n l y t h e p a r t Δ I B C .$ $d r a w Q N / / B R$ $N R = I N = \frac{I R}{2} = \frac{R C}{2}$ $\Rightarrow Q D = \frac{D C}{2}$ $Δ Q D B = \frac{Δ D C B}{2} = \frac{Δ Q C B}{3} = \frac{Δ I B C}{6}$ $Δ R B C = \frac{Δ I B C}{2}$ $A_{I Q D R} = Δ I B C - Δ R B C - Δ Q D B$ $A_{I Q D R} = Δ I B C - \frac{Δ I B C}{2} - \frac{Δ I B C}{6} = \frac{Δ I B C}{3}$ $Σ A_{I Q D R} = \frac{Σ Δ I B C}{3}$ $\Rightarrow h e x a g o n = \frac{Δ A B C}{3}$
	Commented by ajfour last updated on 07/Feb/20

	$E l e m e n t a r y a n d n i c e S i r .$
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