If the circumcentre of a triangle with vertices (x1,y1),(x2,y2),(x3,y3) is (0,0) then prove that the orthocentre is (x1+x2+x3,y1+y2+y3)
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Answer:
OA
2
=OB
2
=OC
2
x
1
2
+x
1
2
tan
2
θ
1
=x
2
2
+x
2
2
tan
2
θ
2
=x
3
2
+x
3
2
tan
2
θ
3
=r
2
x
1
=rcosθ
1
,x
2
=rcosθ
2
,x
3
=rcosθ
3
∴ Co-ordinate of vertices of the triangle become
A(rcosθ
1
,rsinθ
1
),B(rcosθ
2
,rsinθ
2
),C(rcosθ
3
,rsinθ
3
)
x
′
=
3
∑rcosθ
1
,y
′
=
3
∑rsinθ
1
Now, x
′
=
3
x
x
=r(cosθ
1
+cosθ
2
+cosθ
3
)
y
=r(sinθ
1
+sinθ
2
+sinθ
3
)
∴
y
x
=
sinθ
1
+sinθ
2
+sinθ
3
cosθ
1
+cosθ
2
+cosθ
3
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