Home
Class 12
MATHS
AD, BE and CF are the medians of triangl...

AD, BE and CF are the medians of triangle ABC whose centroid is G. If the points A, F, G and E are concyclic, then prove that `2a^(2) = b^(2) + c^(2)`

Text Solution

Verified by Experts

Points A, F, G and E are concylic,

`rArr BG. BE = BF.BA`
`rArr (2)/(3) (BE)^(2) = (1)/(2) c^(2)`
`rArr (2)/(3) xx (1)/(4) (2a^(2) + 2c^(2) -b^(2)) = (1)/(2) c^(2)` (Using Apollonius Theorem)
`rArr 2a^(2) = b^(2) + c^(2)`
Promotional Banner

Topper's Solved these Questions

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE PUBLICATION|Exercise Concept application exercise 5.10|8 Videos

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE PUBLICATION|Exercise Concept application exercise 5.11|4 Videos

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE PUBLICATION|Exercise Concept application exercise 5.8|7 Videos

  • PROGRESSION AND SERIES

    CENGAGE PUBLICATION|Exercise ARCHIVES (NUMERICAL VALUE TYPE )|8 Videos

  • RELATIONS AND FUNCTIONS

    CENGAGE PUBLICATION|Exercise All Questions|1119 Videos

Similar Questions

Explore conceptually related problems

BE and CF are perpendicular on sides AC and AB of triangles ABC respectively . Prove that four points B , C ,E , F are concyclic. Also prove that the two angles of each of DeltaAEFandDeltaABC are equal.

In Delta ABC,/_A = right angle. If CD is a median, then prove that BC^(2) = CD^(2)+ 3AD^(2) .

D , E ,a n dF are the middle points of the sides of the triangle A B C , then (a) centroid of the triangle D E F is the same as that of A B C (b) orthocenter of the tirangle D E F is the circumcentre of A B C

The median AD of the triangle ABC is bisected at E and BE meets AC at F. Find AF : FC.

In the given figure, ABC is a triangle right angled at B. D and E are ponts on BC trisect it. Prove that 8AE^(2) = 3AC^(2) + 5AD^(2) .

(ix) AD, BE and CF are the three medians of Delta ABC and intersect at G. The area of the Delta ABC is 36 Sq. cm. Find (a) the area of Delta AGB and (b) the area of the quadrilateral BDGF.

The medians AD and BE of a triangle ABC with vertices A(0, b), B(0, 0) and C(a, 0) are perpendicular to each other if a^2:b^2 is

Let f, g and h be the lengths of the perpendiculars from the circumcenter of Delta ABC on the sides a, b, and c, respectively. Prove that (a)/(f) + (b)/(g) + (c)/(h) = (1)/(4) (abc)/(fgh)

If the angles A,B,C of a triangle are in A.P. and sides a,b,c, are in G.P., then prove that a^2, b^2,c^2 are in A.P.