Home
Class 12
MATHS
In DeltaABC, the bisectors of the angles...

In `DeltaABC`, the bisectors of the angles A, B and C are extended to intersect the circumcircle at D,E and F respectively. Prove that `AD cos.(A)/(2) + BE cos.(B)/(2) + CF cos.(C)/(2) = 2R (sin A + sin B + sin C)`

Text Solution

Verified by Experts

Using Sine law in `Delta ABD`,
we get

`(AD)/(sin(B + (A)/(2))) = 2R`
`:. AD = 2R sin (B + (A)/(2))`
`rArr AD cos.(A)/(2) = 2R cos.(A)/(2) sin (B + (A)/(2))`
`=R (sin (A + B) + sin B)`
`=R(sin C + sin B)`
Similarly, `BE cos.(B)/(2) = R (sin A + sin C)`
And `CF cos.(C)/(2) = R (sin A + sin B)`
`:. AD cos.(A)/(2) + BE cos.(B)/(2) + CF cos.(C)/(2)`
`=2R(sin A + sin B + sinC)`
Promotional Banner

Topper's Solved these Questions

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE|Exercise Exercise 5.1|12 Videos

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE|Exercise Exercise 5.2|8 Videos

  • PROGRESSION AND SERIES

    CENGAGE|Exercise ARCHIVES (MATRIX MATCH TYPE )|1 Videos

  • PROPERTIES OF TRIANGLE, HEIGHT AND DISTANCE

    CENGAGE|Exercise Question Bank|32 Videos

Similar Questions

Explore conceptually related problems

If A+B+C=180^0 , prove that : cos^2( A/2) + cos^2( B/2) + cos^2(C/2) = 2+2 sin(A/2) sin( B/2) sin( C/2)

(cos A)/(sin B sin C)+(cos B)/(sin C sin A)+(cos C)/(sin A sin B)=2

a.cos A + b.cos B + c.cos C = 2a sin B sin C

a cos A + b cos B + c cos C = 2a sin B sin C

Prove that a cos A+b cos B+c cos C=4R sin A sin B sin C

cos A + cos B + cos C = 1 + 4sin ((A) / (2)) sin ((B) / (2)) sin ((C) / (2))

If A+B+C=180^(@), then prove that cos^(2)(A)/(2)+cos^(2)(B)/(2)+cos^(2)(C)/(2)=2(1+sin(A)/(2)sin(B)/(2)sin(C)/(2))

cos2B + cos2A-cos2C = 1-4sin A sin B cos C

Prove that: sin(B-C)cos(A-D)+sin(C-A)cos(B-D)+sin(A-B)cos(C-D)

(2sin (AC) cos C-sin (A-2C)) / (2sin (BC) cos C-sin (B-2C)) = (sin A) / (sin B)