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Thread: inequality of tringle median length and addition of edges

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    inequality of tringle median length and addition of edges

    How can I prove geometrically: If AM is median of triangle ABC, then: AM < (AB+BC)/2
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    Re: inequality of tringle median length and addition of edges

    Quote Originally Posted by mghorba View Post
    How can I prove geometrically: If AM is median of triangle ABC, then: AM < (AB+BC)/2
    Draw a triangle with vertices A, B and C. A median of a triangle is a line segment connecting a vertex to the midpoint of the opposing side. As such, each triangle has three medians.

    Let's say M is the midpoint on AC. Naturally, 2AM=AC. We also know that the sum of the lengths of any two sides of a triangle is (always) strictly greater than the length of the third side (this applies to all triangles!). Thus,  AC=2AM<AB+BC\implies  AM<\dfrac{AB+BC}{2}.
    Last edited by abender; Jul 7th 2017 at 09:53 AM. Reason: Attending to precision. "Length of side" is better than "side."
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