Show that r+r₁ +r₂ r-₁ =4Rcos C.
Answers
Answer:
r
1
=4Rsin
2
A
cos
2
B
cos
2
C
r
2
=4Rsin
2
B
cos
2
C
cos
2
A
r
3
=4Rsin
2
C
cos
2
A
cos
2
B
r=4Rsin
2
A
sin
2
B
sin
2
C
Now,
r
1
+r
2
−r
3
+r=4R[4Rsin
2
A
cos
2
B
cos
2
C
+sin
2
B
cos
2
C
cos
2
A
−sin
2
C
cos
2
A
cos
2
B
+sin
2
A
sin
2
B
sin
2
C
]
=4R[cos
2
C
(sin
2
A
cos
2
B
+sin
2
B
cos
2
A
)−sin
2
C
(cos
2
A
cos
2
B
−sin
2
A
sin
2
B
)]
=4R[cos
2
C
sin
2
A+B
−sin
2
C
cos
2
A+B
]=4R[cos
2
C
sin(
2
π
−
2
C
)−sin
2
C
cos(
2
π
−
2
C
)]
=4R[cos
2
C
cos
2
C
−sin
2
C
sin
2
C
]=4RcosC
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SIMILAR QUESTIONS
star-struck
cos11
o
−cos2
o
> 0
Medium
View solution
>
If I,O and P be respectively the incentre, circumcentre and orthocentre and G the centroid of the triangle ABC prove that
IP
2
=2r
2
−4R
2
cosAcosBcosC.
Step-by-step explanation:
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