Prove medians of a triangle can make a triangle

Question

Prove medians of a triangle can make a triangle. It means: If medians are: $m_a,m_b,m_c$, then we have $m_a + m_b > m_c$.

I know we can prove it using the length of medians (Apollonius theorem) but I want a geometric prove, not using pure algebra and square roots and similar things.

Thanks and sorry for my English.

Answer 1

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Answer 2

With $z \in \mathbb{C}$ being the complex number associated with point $Z$ in the complex plane $2 \cdot m_a=|b+c-2a|$ and similar for $m_b,m_c$, so the inequality to prove becomes: $$|b+c-2a| + |c+a-2b| \ge |a+b-2c|$$ which follows by the triangle inequality $|z_1| + |z_2| \ge |z_1+z_2|$ for $\;z_1=b+c-2a$,$\;z_2=c+a-2b$,$\;z_1+z_2=2c-a-b$.