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Question Number 206922 by mr W last updated on 01/May/24

s o l v e f o r x, y, z \in R^{+}

x^{2} + y^{2} - 2 x y \cos γ = c^{2}

y^{2} + z^{2} - 2 y z \cos α = a^{2}

z^{2} + x^{2} - 2 z x \cos β = b^{2}

w i t h α + β + γ = 360 °

e x a m p l e :

a = 12, b = 8, c = 10

α = 120 °, β = 90 °, γ = 150 °

Answered by mr W last updated on 01/May/24

Commented by mr W last updated on 01/May/24

Commented by mr W last updated on 01/May/24

∠ A = \cos^{- 1} \frac{b^{2} + c^{2} - a^{2}}{2 b c}

∠ B = \cos^{- 1} \frac{c^{2} + a^{2} - b^{2}}{2 c a}

∠ C = \cos^{- 1} \frac{a^{2} + b^{2} - c^{2}}{2 a b}

π - 2 ϕ_{1} = 2 π - 2 α

\Rightarrow ϕ_{1} = α - \frac{π}{2}

s i m i l a r l y

ϕ_{2} = β - \frac{π}{2}

ϕ_{3} = γ - \frac{π}{2}

r_{1} = \frac{a}{2 \cos ϕ_{1}} = \frac{a}{2 \sin α}

s i m i l a r l y

r_{2} = \frac{b}{2 \sin β}

r_{3} = \frac{c}{2 \sin γ}

{Q R}^{2} = r_{2}^{2} + r_{3}^{2} - 2 r_{2} r_{3} \cos (∠ A + ϕ_{2} + ϕ_{3})

∠ A + ϕ_{2} + ϕ_{3} = ∠ A + β + γ - π = π - (α - ∠ A)

{Q R}^{2} = \frac{b^{2}}{4 \sin^{2} β} + \frac{c^{2}}{4 \sin^{2} γ} + \frac{b c \cos (α - ∠ A)}{2 \sin β \sin γ}

\frac{x \times Q R}{2} = r_{2} r_{3} \sin (∠ A + ϕ_{2} + ϕ_{3}) = 2 [A Q R]

\frac{x \times Q R}{2} = \frac{b c \sin (α - ∠ A)}{4 \sin β \sin γ}

\Rightarrow x = \frac{b c \sin (α - ∠ A)}{\sin β \sin γ \sqrt{\frac{b^{2}}{\sin^{2} β} + \frac{c^{2}}{\sin^{2} γ} + \frac{2 b c \cos (α - ∠ A)}{\sin β \sin γ}}}

s i m i l a r l y

\Rightarrow y = \frac{c a \sin (β - ∠ B)}{\sin γ \sin α \sqrt{\frac{c^{2}}{\sin^{2} γ} + \frac{a^{2}}{\sin^{2} α} + \frac{2 c a \cos (β - ∠ B)}{\sin γ \sin a}}}

\Rightarrow z = \frac{a b \sin (γ - ∠ C)}{\sin α \sin β \sqrt{\frac{a^{2}}{\sin^{2} α} + \frac{b^{2}}{\sin^{2} β} + \frac{2 a b \cos (γ - ∠ C)}{\sin α \sin β}}}

e x a m p l e :

a = 12, b = 8, c = 10

α = 120 °, β = 90 °, γ = 150 °

\cos ∠ C = \frac{12^{2} + 8^{2} - 10^{2}}{2 \times 12 \times 8} = \frac{9}{16}

\sin (γ - ∠ C) = \frac{9 + 5 \sqrt{21}}{32}

\cos (γ - ∠ C) = \frac{- 9 \sqrt{3} + 5 \sqrt{7}}{32}

z = \frac{12 \times 8 \times \frac{9 + 5 \sqrt{21}}{32}}{\frac{\sqrt{3}}{2} \sqrt{\frac{12^{2} \times 4}{3} + 8^{2} + \frac{2 \times 12 \times 8 \times 2}{\sqrt{3}} \times \frac{- 9 \sqrt{3} + 5 \sqrt{7}}{32}}}

= \frac{9 \sqrt{3} + 15 \sqrt{7}}{\sqrt{37 + 5 \sqrt{21}}} \approx 7.141

Commented by mr W last updated on 01/May/24

g e o m e t r i c a l l y i t i s t o f i n d t h e

d i s t a n c e s f r o m a p o i n t i n s i d e a

t r i a n g l e t o t h e v e r t e x e s o f t h e t r i a n g l e .

w h e n α = β = γ = 120 °, t h i s p o i n t i s

t h e s o - c a l l e d f e r m a t p o i n t .

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