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Question Number 206269 by cortano21 last updated on 10/Apr/24

Commented by A5T last updated on 11/Apr/24

Commented by A5T last updated on 11/Apr/24

$G e n e r a l f o r m u l a . W h e n k_{1} k_{2} k_{3} = 1, {C e v a}^{'} s$ $t h e o r e m f o l l o w s s i n c e [P_{1} P_{2} P_{3}] = 0 .$
	Answered by TonyCWX08 last updated on 11/Apr/24

	$U s i n g {H e r o n}^{'} s F o r m u l a :$ $S_{A B C} = 15 \sqrt{510}$ $U s i n g {R o u t h}^{'} s T h e o r e m :$ $S_{R G T} = \frac{{(7 \times 7 \times 5)}^{2} (15 \sqrt{510})}{(7 (7) + 7 + 1) (7 (5) + 5 + 1) (5 (7) + 7 + 1)}$ $S_{R G T} = \frac{60025 (15 \sqrt{510})}{(57) (41) (43)}$ $S_{R G T} = \frac{60025 (15 \sqrt{510})}{100491}$ $S_{R G T} = \frac{60025 (15 \sqrt{510})}{100491}$ $S_{R G T} = \frac{300125 \sqrt{510}}{33497}$ $F i n a l l y :$ $\frac{S_{R G T}}{S_{A B C}} = \frac{300125 \sqrt{510}}{33497} \times \frac{1}{15 \sqrt{510}} = \frac{60025}{100491} \approx 0.6$
	Commented by TonyCWX08 last updated on 11/Apr/24

	$P l e a s e C h e c k .$ $I^{'} m a b i t u n s u r e .$
	Commented by A5T last updated on 11/Apr/24

	$I g u e s s t h e f o r m u l a a b o v e c o u l d b e t h e c o r r e c t$ $a p p l i c a t i o n o f {R o u t h}^{'} s t h e o r e m .$
	Answered by mr W last updated on 11/Apr/24

	$\frac{M G}{G B} \times \frac{7}{3} \times \frac{8}{1} = 1 \Rightarrow \frac{M G}{G B} = \frac{3}{56}$ $\Rightarrow [A M G] = \frac{3}{59} [A M B] = \frac{3}{59} \times \frac{1}{8} [A B C]$ $\frac{Q R}{R C} \times \frac{7}{1} \times \frac{7}{2} = 1 \Rightarrow \frac{Q R}{R C} = \frac{2}{49}$ $\Rightarrow [B Q R] = \frac{2}{51} [B Q C] = \frac{2}{51} \times \frac{2}{7} [A B C]$ $\frac{N T}{T A} \times \frac{5}{2} \times \frac{10}{3} = 1 \Rightarrow \frac{N T}{T A} = \frac{3}{25}$ $\Rightarrow [C N T] = \frac{3}{28} [C N A] = \frac{3}{28} \times \frac{3}{10} [A B C]$ $[R G T] = [A B C] - [A M B] - [B Q C] - [C N A] + [A M G] + [B Q R] + [C N T]$ $[R G T] = [A B C] - \frac{1}{8} [A B C] - \frac{2}{7} [A B C] - \frac{3}{10} [A B C] + \frac{3}{59} \times \frac{1}{8} [A B C] + \frac{2}{51} \times \frac{2}{7} [A B C] + \frac{3}{28} \times \frac{3}{10} [A B C]$ $\frac{[R G T]}{[A B C]} = 1 - \frac{1}{8} - \frac{2}{7} - \frac{3}{10} + \frac{3}{59} \times \frac{1}{8} + \frac{2}{51} \times \frac{2}{7} + \frac{3}{28} \times \frac{3}{10}$ $= \frac{57121}{168504} \approx 0.339$
	Commented by TonyCWX08 last updated on 11/Apr/24

	$W h y e q u a l t o 1 ?$
	Commented by cortano21 last updated on 11/Apr/24

	$M e n e l o u s t h e o r e m$
	Commented by mr W last updated on 11/Apr/24

	Answered by A5T last updated on 11/Apr/24

	$[B Q C] = \frac{2}{7} [A B C] = [B Q R] + [B R T N] + [T N C]$ $[B A M] = \frac{1}{8} [A B C] = [B Q R] + [Q R G A] + [A G M]$ $[A N C] = \frac{3}{10} [A B C] = [A G M] + [M G T C] + [T N C]$ $\Rightarrow [A B C] (\frac{2}{7} + \frac{1}{8} + \frac{3}{10}) = [A B C] - [G R T] + [B Q R]$ $+ [T N C] + [A G M] . . . (i)$ $\frac{Q R}{R C} \times \frac{7}{1} \times \frac{7}{2} = 1 \Rightarrow \frac{Q R}{R C} = \frac{2}{49} \Rightarrow [B Q R] = \frac{4 [A B C]}{51 \times 7}$ $\frac{7}{3} \times \frac{8}{1} \times \frac{M G}{G B} = 1 \Rightarrow \frac{M G}{G B} = \frac{3}{56} \Rightarrow [A M G] = \frac{3 [A B C]}{472}$ $\frac{5}{2} \times \frac{10}{3} \times \frac{N T}{T A} = 1 \Rightarrow \frac{N T}{T A} = \frac{3}{25} \Rightarrow [C N T] = \frac{9}{280} [A B C]$ $(i) \Rightarrow \frac{[G R T]}{[A B C]} = 1 - \frac{2}{7} - \frac{1}{8} - \frac{3}{10} + \frac{4}{357} + \frac{3}{472} + \frac{9}{280}$ $= \frac{57121}{168504} \approx 0.339$
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