# Geometry: Triangle similarity within another triangle

Assume a $\Delta SRQ$, where point N is the midpoint of SR and point M is the midpoint of SQ. Lines MR and NQ intersect at point P.

Is $\Delta MNP$ similar to $\Delta RQP$ ? I know that $\angle MPN$ and $\angle QPR$ are congruent, and that sides MN and QR are proportional, but that's only SA ... am I missing another side?

Thanks for the help!

## 2 Answers

The triangles are similar. You can actually prove that the bisection creates four similar triangles:

$SN = \frac{SR}{2}$ $SM = \frac{SQ}{2}$

So you can prove similarity by side-angle-side. You can do the same for the other three angles.

After you've proven that, you now have proven that $MN = \frac{RQ}{2}$, etc. So the fourth triangle is proven similar by side-side-side.

$MR$ and $NQ$ are medians of the triangle, and it is easily provable that medians intersect each other in $1/3$ of their lengths, so the parts are in $1:2$ ratio — the same as $MN : QR$. And here you have sides proportional, which implies triangles' similarity.