Home
Class 12
MATHS
If AD, BE, CF are the diameters of circu...

If AD, BE, CF are the diameters of circumcircle of `Delta ABC`, then prove that area of hexagon AFBDCE is `2Delta`

Text Solution

Verified by Experts

Chord AC subtends the same angle at point B and D
`:. Angle ADC = B`
So, in right angled triangled ACD
`CD = AC cot B`
`:. CD = b cot B`
Similarly from `Delta ABD`,
`BD = c cot C`

`:. " Area of " Delta BDC`
`Delta_(BDC) = (1)/(2) (b cot B) (c cot C) sin (B + C)`
`= (bc)/(2) sin A cot C cot B`
`= (abc)/(4R) cot B cot C`
Similarly `Delta_(AEC) = (abc)/(4R) cot A cot C`
and `Delta_(AFB) = (abc)/(4R) cot A cot B`
`:.` Area of hexagon AFBDCE
`= Delta_(BDC) + Delta_(AFC) + Delta_(AFB) + Delta_(ABC)`
`= (abc)/(4R) [cot A cot B + cot B cot C + cot C cot A] + Delta`
`= Delta + Delta`
(as in `Delta ABC, cot A cot B + cot B cot C + cot A = 1`)
`=2 Delta`
Promotional Banner

Topper's Solved these Questions

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE|Exercise Exercise 5.8|7 Videos

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE|Exercise Exercise 5.9|5 Videos

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE|Exercise Exercise 5.6|6 Videos

  • PROGRESSION AND SERIES

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

  • PROPERTIES OF TRIANGLE, HEIGHT AND DISTANCE

    CENGAGE|Exercise Question Bank|15 Videos

Similar Questions

Explore conceptually related problems

D, E, F are mid points of sides BC, CA, AB of Delta ABC . Find the ratio of areas of Delta DEF and Delta ABC .

Let the lengths of the altitudes drawn from the vertices of Delta ABC to the opposite sides are 2, 2 and 3. If the area of Delta ABC " is " Delta , then find the area of triangle

If AD, BE and CF are the altitudes of Delta ABC whose vertex A is (-4,5). The coordinates of points E and F are (4,1) and (-1,-4), respectively. Equation of BC is

Find the lengths of chords of the circumcircle of triangle ABC, made by its altitudes

If veca,vecb,vecc are position vectors of the vertices A,B,C of a triangle ABC, show that the area of the triangle ABC is (1)/(2) |vecaxxvecb+vecbxxvecc+veccxxveca| . Also deduce the condition for collinearity of the points A,B and C.

In triangle A B C ,/_A=60^0,/_B=40^0,a n d/_C=80^0dot If P is the center of the circumcircle of triangle A B C with radius unity, then the radius of the circumcircle of triangle B P C is (a)1 (b) sqrt(3) (c) 2 (d) sqrt(3) 2

ABC is a right angled triangle with angle B = 90 ^@ , a=6 cm if the radius of the circumcircle is 5 cm then the area of Delta ABC is

Construct a circumcircle of the triangle ABC where AB = 5cm, angleB = 75^@ and BC = 7cm

Draw the circumcircle for an equilateral triangle of side 6 cm.

If the perimeter of the triangle formed by feet of altitudes of the triangle ABC is equal to four times the circumradius of Delta ABC , then identify the type of Delta ABC