The Medians, Midpoints, and Medial Triangle

In $\Delta ABC$, the medial triangle is defined as the triangle with vertices at the midpoints of the sidelengths of $\Delta ABC.$

One useful property is that the medial triangle is similar to the original one.

PROOF OF SIMILARITY:

Consider $\Delta ABC$. Denote the midpoints of $AB$ as $F$, $BC$ as $D$ and $AC$ as $E.$ Also, express $AB$ as $c$, $BC$ as $a$, and $AC$ as $b.$ And finally, we have $AF = BF = \frac{c}{2}$, $AE = CE = \frac{b}{2}$, and $BD = CD = \frac{a}{2}.$

Consider $\Delta AFE:$

We have $AF = \frac{c}{2}$ and $AE = \frac{b}{2}.$

Since $$\frac{c}{c/2} = \frac{b}{b/2}$$

and $\Delta ABC \sim \Delta AFE$,

$FE = \frac{a}{2}.$

Similarly $ED = \frac{c}{2}$ and $FD = \frac{b}{2}.$

Hence, $\Delta ABC \sim \Delta DEF.$

$\blacksquare$

CONCLUSION:

These triangles are very common so it's useful to know about them and their properties.