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Question Number 61521 by necx1 last updated on 03/Jun/19

	Answered by ajfour last updated on 04/Jun/19

	Commented by ajfour last updated on 04/Jun/19

	$F B l i n e = \frac{b}{4} + λ (a - \frac{b}{4})$ $B E l i n e = μ (\frac{a}{4} + \frac{3 b}{4})$ $r_{P} = \frac{b}{4} + λ (a - \frac{b}{4}) = μ (\frac{a}{4} + \frac{3 b}{4})$ $\Rightarrow 4 λ = μ & 1 - λ = 3 μ$ $\Rightarrow λ = \frac{1}{13}, μ = \frac{4}{13}$ $\Rightarrow r_{P} = B P = \frac{a + 3 b}{13} . . . . . . . (i)$ $A D = b + ρ (\frac{3 a}{7} - b)$ $r_{Q} = \frac{b}{4} + λ_{1} (a - \frac{b}{4}) = b + ρ (\frac{3 a}{7} - b)$ $\Rightarrow 4 - 4 ρ = 1 - λ_{1}$ $a n d λ_{1} = \frac{3 ρ}{7}$ $\Rightarrow ρ = \frac{21}{25}, λ_{1} = \frac{9}{25}$ $r_{Q} = \frac{4 b + 9 a}{25} . . . . . (i i)$ $r_{R} = μ_{1} (\frac{a}{4} + \frac{3 b}{4}) = b + ρ_{1} (\frac{3 a}{7} - b)$ $\Rightarrow μ_{1} = \frac{12}{7} ρ_{1}; 4 - 4 ρ_{1} = 3 μ_{1}$ $\Rightarrow 12 - 7 μ_{1} = 9 μ_{1}$ $\Rightarrow μ_{1} = \frac{3}{4}, ρ_{1} = \frac{7}{16}$ $r_{R} = \frac{3 a}{16} + \frac{9 b}{16} . . . . (i i i)$ $P Q = r_{Q} - r_{P} = \frac{4 b + 9 a}{25} - \frac{a + 3 b}{13}$ $P R = r_{R} - r_{P} = \frac{3 a + 9 b}{16} - \frac{a + 3 b}{13}$ $A_{s h a d e d} = \frac{1}{2} ∣ P Q \times P R ∣$ $= \frac{1}{2} [(\frac{9}{25} - \frac{1}{13}) (\frac{9}{16} - \frac{3}{13}) - (\frac{4}{25} - \frac{3}{13}) (\frac{3}{16} - \frac{1}{13})] ∣ a \times b ∣$ $= [\frac{92}{25 \times 13} \times \frac{69}{16 \times 13} + \frac{23}{25 \times 13} \times \frac{23}{16 \times 13}] △_{A B C}$ $= [\frac{92 \times 69 + 23 \times 23}{25 \times 16 \times 169}] △_{A B C}$ $A_{s h a d e d} = \frac{529}{5200} △_{A B C} \approx 10 . 173 % .$ $△_{A B C} = \sqrt{\frac{19}{2} (\frac{19}{2} - 7) (\frac{19}{2} - 4) (\frac{19}{2} - 8)}$ $= \frac{\sqrt{19 \times 5 \times 11 \times 3}}{4}$ ${A r e a}_{s h a d e d} = \frac{529}{5200} \times \frac{\sqrt{3135}}{4}$ $A_{s h a d e d} = \frac{529 \sqrt{3135}}{20800} \approx 1.424004$
	Commented by Tawa1 last updated on 05/Jun/19

	$God bless you sir$
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