Prove that triangle midline and median split themselves into halves

Question

How can I prove that median and midline in a triangle split themselves into halves?

Thanks a lot in advance!

Answer 1

HINT.

• In a triangle a midline is parallel to the third side and is a half of that.

• In a triangle a line through the midpoint of a side and parallel to another side, cuts the third side at its midpoint.

Answer 2

Let $\;\Delta ABC\;$ be the triangle, $\;EF\;$ the midline joining the middle points of $\;AB,\,AC\;$ and $\;AM\;$ the median to $\;BC\;$, and let $\;K\;$ be the point of intersection of $\;EF\;$ and $\;AM\;$ .

Since $\;EF||BC\;$, Thales (the intercepts theorems for parallels) theorem tells us, applied twice, that

$$\text{On}\;\;\Delta ABM\;:\;\frac{AE}{EB}=\frac{AK}{KM}=\color{red}{\frac{EK}{BM}}\;,\;\;\;\;\;\;\;\text{On}\;\Delta AMC\;:\frac{AK}{KM}=\frac{AF}{FC}=\color{red}{\frac{KF}{MC}} $$

Now just equate both red parts (which are equal!) and remember that $\;BM=MC\;$ ...