a, b, c are the lengths of three sides of a triangle ABC. If a, b, c are related by the relation a² + b² + c² = ab + bc + ca, then the value of sin²A+ sin²B + sin²C is

$3 over 4$
$fraction numerator 3 square root of 3 over denominator 2 end fraction$
$3 over 2$
$3 over 2$

3 over 2

straight a squared plus straight b squared plus straight c squared space equals space ab space plus space bc space plus space ca rightwards double arrow space space 2 straight a squared space plus space 2 straight b squared space plus space 2 straight c squared space space equals space 2 ab space plus space 2 bc space plus space 2 ca rightwards double arrow space straight a squared plus space straight a squared space plus space straight b squared space plus space straight b squared space plus space straight c squared space plus space straight c squared space minus space 2 ab space minus space 2 bc minus space 2 ca space equals space 0

rightwards double arrow space space straight a squared plus straight b squared minus 2 ab space plus space straight b squared plus straight c squared minus 2 bc space plus space straight c squared space plus space straight a squared space minus space 2 ca space equals space 0

rightwards double arrow space left parenthesis straight a minus straight b right parenthesis squared space plus space left parenthesis straight b minus straight c right parenthesis squared space plus space left parenthesis straight c minus straight a right parenthesis squared space equals space 0 rightwards double arrow space space space straight a space minus space straight b space equals space 0 space space space rightwards double arrow space space straight a space equals space straight b

straight b minus straight c space equals space 0 space space space rightwards double arrow space space straight b space equals space straight c straight c space minus space straight a space equals space 0 space rightwards double arrow space straight c space equals space straight a

Δ ABC is an equilateral triangle.

therefore space space angle straight A space equals space angle straight B space equals space angle straight C space equals space 60 degree therefore space sin squared straight A space plus space sin squared straight B space plus space sin squared straight C

equals space 3 sin squared straight A space plus space sin squared straight B space plus space sin squared straight C equals space 3 space sin squared straight A space space equals space 3 space cross times space sin squared 60 degree equals space 3 space cross times space open parentheses fraction numerator square root of 3 over denominator 2 end fraction close parentheses squared equals space fraction numerator 3 space cross times space 3 over denominator 4 end fraction space equals space 9 over 4

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