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Congruency of Triangles

If a,b,c are ...

Question

If a,b,c are all non zero and a+b+c=0,prove that

a²/bc + b²/ca + c²/ab =3

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Solution

$fraction numerator a squared over denominator b c end fraction plus fraction numerator b squared over denominator c a end fraction plus fraction numerator c squared over denominator a b end fraction equals fraction numerator a cubed plus b cubed plus c cubed over denominator a b c end fraction minus negative negative negative negative negative negative left parenthesis 1 right parenthesis a cubed plus b cubed plus c cubed minus 3 a b c equals left parenthesis a plus b plus c right parenthesis left parenthesis a squared plus b squared plus c squared minus a b minus b c minus c a right parenthesis W h e n space a plus b plus c equals 0 comma a cubed plus b cubed plus c cubed minus 3 a b c equals 0 s o space a cubed plus b cubed plus c cubed equals 3 a b c S u b s t i t u t i n g space i n space left parenthesis 1 right parenthesis space w e space g e t fraction numerator 3 a b c over denominator a b c end fraction equals 3 minus negative left parenthesis p r o v e d right parenthesis$

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Q. Question 7
If a, b and c are all non-zero and a + b + c = 0, then prove that

\frac{a^{2}}{b c} + \frac{b^{2}}{a c} + \frac{c^{2}}{a b} = 3

If a, b, c are rational numbers such that a² + b² + c² − ab − bc − ca = 0, prove that a = b = c.

Q. Prove that a² + b² + c² −ab−bc−ca is always non-negative for all values of a, b, and c.

Q. If a², b², c² are in A.P., prove that

\frac{a}{b + c}, \frac{b}{c + a}, \frac{c}{a + b}

a + b + c

are all non-zero &

a + b + c = 0

\frac{a^{2}}{b c} + \frac{b^{2}}{c a} + \frac{c^{2}}{a b} = 3

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Congruency of Triangles

Standard IX Mathematics

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