posted by .

If a, b and c are sides of a triangle
Then prove that 1≤ [(a^2+b^2+c^2)/(ab + bc+ca)]

  • maths -

    so, we want to show that

    ab+bc+ca <= (a^2+b^2+c^2)

    since all squares are positive,

    0 <= (a-b)^2 + (a-c)^2 + (b-c)^2
    0 <= a^2-2ab+b^2 + a^2-2ac+c^2 + b^2-2bc+c^2
    0 <= 2(a^2+b^2+c^2) - 2(ab+ac+bc)
    ab+ac+bc <= a^2+b^2+c^2

    This is in fact true for any three real numbers, not just sides of a triangle

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. Maths

    X and Y are points on the sides BC and AC of a triangle ABC respectively such that angle AXC = angle BYC and BX = XY. Prove that AX bisects the angle BAC. I believe the first step is to prove ABXY is a cyclic quadrilateral. Unfortunately, …
  2. Maths

    Prove that any two sides of a triangle are together greater than twice the median drawn to third side.
  3. Maths

    Prove that the triangle formed by the points of contact of the sides of a given triangle with the excircles corresponding to those sides is equivalent to the triangle formed by the points of contact of the sides of the traingle with …
  4. algebra 1 help please

    4) a student score is 83 and 91 on her first two quizzes. write and solve a compound inequality to find possible values for a thord quiz score that would give anverage between 85 and 90. a. 85≤83+91+n/3 ≤90; 81≤n≤96 …
  5. MATH Calculus Homework PLEASE HELP!

    Let T be the triangle 1≤x≤2, 0≤y≤3x-3. Find the volume that results by rotating the triangle T around the line x=-1.
  6. Maths(help me plz..!

    The sides of a triangle are of lenths x*-y*, x*+y* and 2xy units respectively. Prove that the triangle is a right angled triangle
  7. Maths

    Let f be a function such that f(−6)=−6, f(6)=6, f is differentiable for all real values of x and −1≤f'x≤1 for all real values of x. Prove that f(x)=x for all −6≤x≤6 I tried applying the …

    Eliminate the parameter t. Find a rectangular equation for the plane curve defined by the parametric equations. x = 6 cos t, y = 6 sin t; 0 ≤ t ≤ 2π A. x2 - y2 = 6; -6 ≤ x ≤ 6 B. x2 - y2 = 36; -6 ≤ …
  9. Maths

    ABC is an isosceles triangle in which the equal sides are AB and A.If AD is an altitude and |BC|=1/2|AB|.Prove that |AD|^2=15|BC|^2
  10. bctc

    Suppose that 3 ≤ f '(x) ≤ 5 for all values of x. prove that 18 ≤ f(8)-f(2) ≤ 30 ?

More Similar Questions