# Prove medians of a triangle can make a triangle

Prove medians of a triangle can make a triangle.
It means:
If medians are: ma,mb,mcm_a,m_b,m_c, then we have ma+mb>mcm_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.

=================

1

Note that a vector proof (which may be too “algebraic”) is immediate: With DD, EE, FF the midpoints opposite AA, BB, CC, we have →AD=D−A=12(−2A+2B+2C)→BE=E−B=12(−2A−2B+2C)→CF=F−C=12(−2A+2B−2C)\begin{align}\overrightarrow{AD} &= D-A = \frac{1}{2}(-2A+\phantom{2}B+\phantom{2}C) \\ \overrightarrow{BE} &= E – B = \frac{1}{2}(\phantom{-2}A-2B+\phantom{2}C) \\ \overrightarrow{CF} &= F – C = \frac{1}{2}(\phantom{-2}A +\phantom{2}B – 2 C)\end{align} so that →AD+→BE+→CF=0◻\overrightarrow{AD} + \overrightarrow{BE} + \overrightarrow{CF} = 0 \qquad\square
– Blue
2 days ago

=================