Prove that any two sides of a trianglr are together greater than twice thr median drawn to the third side
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To prove that any two sides of a triangle are together greater than twice the median drawn to the third side, we can use the triangle inequality theorem. The triangle inequality theorem states that for any triangle with sides of lengths a, b, and c, the sum of the lengths of any two sides must be greater than the length of the remaining side.
Let's denote the sides of the triangle as a, b, and c. Without loss of generality, let's assume that c is the longest side. The median drawn to side c splits the triangle into two smaller triangles.
Now, let's consider the two smaller triangles formed by the median:
Triangle 1:
Sides: a, c/2, and median
Triangle 2:
Sides: b, c/2, and median
By the triangle inequality theorem, in Triangle 1, we have:
a + c/2 > median
And in Triangle 2, we have:
b + c/2 > median
Adding these two inequalities together, we get:
(a + c/2) + (b + c/2) > median + median
(a + b + c)/2 > 2 x median
Therefore, we have proven that the sum of any two sides of a triangle is greater than twice the median drawn to the third side.