Suppose a, b, c are sides of a scalene triangle. Let Δ$$=a$$ b cb c ac a b .Then

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Suppose a, b, c are sides of a scalene triangle. Let Δ$$=a$$    b    cb    c    ac    a    b .Then

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Using $$C1$$→$$C1+C2+C3$$ we getΔ$$=(a+b+c)$$Δ$$1where$$  Δ$$1=1$$    b    $$c1$$    c    $$a1$$    a    bApplying $$R2$$→$$R2$$−$$R1$$,$$R3$$→$$R3$$−$$R1$$ we getΔ$$1=1bc0c$$−ba−$$c0a$$−bb−$$c=$$−(b$$−$$c)2$$−$$(a$$−$$b)(a$$−$$c)=$$−$$a2+b2+c2$$−bc−ca−ab⇒        Δ$$1=$$−$$12(b$$−$$c)2+(c$$−$$a)2+(a$$−$$b)2$$<$$0$$ As $$a+b+c$$>$$0$$, we getΔ$$=(a+b+c)Δ$$1$$<$$0$$

Suppose $a, b, c$ are sides of a scalene triangle. Let $Δ = |\begin{array}{lll} a b c \\ b c a \\ c a b \end{array}|$ .Then

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Suppose a, b, c are sides of a scalene triangle. Let Δ=a b cb c ac a b .Then

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Suppose $a, b, c$ are sides of a scalene triangle. Let $Δ = |\begin{array}{lll} a b c \\ b c a \\ c a b \end{array}|$ .Then