Show that for any triangle with sides `a ,b ,a n dc3(a b+b c+c a)<(a+b+c)^2<4(b c+c a+a b)dot` When are the first two expressions equal ?

If vec xa n d vec y are two non-collinear vectors and A B C isa triangle with side lengths a ,b ,a n dc satisfying (20 a-15 b) vec x+(15b-12 c) vec y+(12 c-20 a)( vecx xx vec y)=0, then triangle A B C is a. an acute-angled triangle b. an obtuse-angled triangle c. a right-angled triangle d. an isosceles triangle

If Delta is the area of a triangle with side lengths a, b, c, then show that as Delta leq 1/4 sqrt((a + b + c) abc) Also, show that the equality occurs in the above inequality if and only if a = b = c .

If Delta is the area of a triangle with side lengths a, b, c, then show that as Delta leq 1/4 sqrt((a + b + c) abc) Also, show that the equality occurs in the above inequality if and only if a = b = c .

Show that in any triangle ABC, (a+b+c) (tan (A/2) + tan (B/2)) = 2c cot (C/2)

If in a triangle A B C ,acos^2(C/2)+ c cos^2(A/2)=(3b)/2, then the sides a ,b ,a n dc are in A.P. b. are in G.P. c. are in H.P. d. satisfy a+b=cdot

If a ,b ,c are sides of a scalene triangle, then value of |[a, b, c] ,[b, c, a] ,[c, a, b]| is positive (b) negative non-positive (d) non-negative

In a triangle, if the angles A , B ,a n dC are in A.P. show that 2cos1/2(A-C)=(a+c)/(sqrt(a^2-a c+c^2))

Let a,b,c denote the lengths of the sides of a triangle such that ( a-b) vecu + ( b-c) vecv + ( c-a) (vecu xx vecv) = vec0 For any two non-collinear vectors vecu and vecu ,then the triangle is

If sides of triangle A B C are a , ba n dc such that 2b=a+c then b/c >2/3 (b) b/c >1/3 b/c<2 (d) b/c<3/2

If D , E ,a n dF are three points on the sides B C ,A C ,a n dA B of a triangle A B C such that A D ,B E ,a n dC F are concurrent, then show that BD*CE*AF=DC*EA*FB .