If a+b+c80, ax +by+c= 0 bx + cy+ a = 0, cx + ay+b=0
a² +6² +6²
are concurrent then ab + bc+ca
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ax+by+c=0
bx+cy+a=0
cx+ay+b=0
Adding all the above equations, we get,
ax+by+bx+cy+a+cx+ay+b=0
ax+ay+a+bx+by+b+cx+xy+c=0
a(x+y+1)+b(x+y+1)+c(x+y+1)=0
(x−y+1)(a+b+c)=0
Assuming (a+b+c)=0, we get,
a+b=−c.............(1)
cubing both sides,
a
3
+b
3
+3ab(a+b)=−c
3
a
3
+b
3
+3ab(−c)=−c
3
.........fromeq.(1)
a
3
+b
3
−3abc=−c
3
a
3
+b
3
+c
3
=3abc
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