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)( vec xxx 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 .

Let C be incircle of A B Cdot If the tangents of lengths t_1,t_2a n dt_3 are drawn inside the given triangle parallel to sidese a , ba n dc , respectively, the (t_1)/a+(t_2)/b+(t_3)/c is equal to 0 (b) 1 (c) 2 (d) 3

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))

A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle. Show that the minimum length of the hypotenuse is (a^((2)/(3))+b^((2)/(3)))^((3)/(2)) .

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

a triangle A B C with fixed base B C , the vertex A moves such that cosB+cosC=4sin^2(A/2)dot If a ,ba n dc , denote the length of the sides of the triangle opposite to the angles A , B ,a n dC , respectively, then (a) b+c=4a (b) b+c=2a (c) the locus of point A is an ellipse (d) the locus of point A is a pair of straight lines

If a, b, c are the sides of a scalene triangle such that |[a , a^2 , a^(3)-1 ],[b , b^2 , b^(3)-1],[ c, c^2 , c^(3)-1]|=0 , then the geometric mean of a, b c is

Consider a DeltaABC in which the sides are a = (n +1), b = (n + 2), c = n with tan C = 4//3 , then the value of Delta is _____