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Byju's Answer
Standard XII
Mathematics
Angle between Two Lines
If P is ortho...
Question
If P is orthocentre, Q is circumcenter and G is centroid of a triangle ABC, then prove that
¯
¯¯¯¯¯¯
¯
Q
P
=
3
¯
¯¯¯¯¯¯¯
¯
Q
G
.
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Solution
Let
¯
p
and
¯
g
be the poistion vectors of
P
and
G
w.r.t. the circumcentre
Q
.
i.e.
¯
¯¯¯¯¯¯¯
¯
Q
R
=
¯
p
and
¯
¯¯¯¯¯¯¯
¯
Q
G
=
¯
g
We know that
Q
,
G
,
P
are collinear and
G
diviced segment
Q
P
internally in the ratio
1
:
2.
∴
by section formula for internal division,
¯
g
=
1.
¯
p
+
2
¯
q
1
+
2
=
¯
p
3
......
[
∵
¯
q
=
0
]
∴
¯
p
=
3
¯
g
¯
¯¯¯¯¯¯
¯
Q
P
=
3
¯
¯¯¯¯¯¯¯
¯
Q
G
.
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Q.
If ABC is a triangle whose orthocentre is P, circumcentre is Q then prove that
¯
¯¯¯¯¯¯
¯
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¯
¯¯¯¯¯¯
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