If you draw the altitude to the right triangle as shown, it is easily seen that $$\triangle KLM\sim\triangle KNL\sim\triangle LNM.$$ This in turn leads to several interesting proportional relations like $$\frac{KN}{KL}=\frac{KL}{KM} \qquad\text{and}\qquad \frac{KN}{LN}=\frac{LN}{NM}.$$ These turn out to be crucial to what is arguably the canonical geometric proof of the Pythagorean theorem.
(Quick version: since $AC^2=AD\cdot AB$ and $BC^2=BD\cdot AB$, the two red regions have the same area and the two green regions have the same area.)
I'm looking for strategies to help my students master this concept. Is there a special name for this key geometric observation that would help my search, like how Thales' theorem was important enough to get its own name? My personal experience, even as a gifted student, is that it can be hard to see the similarity when you have to reflect one of the triangles to visualize the dilation compared to a similarity diagram with a triangle and one of its midsegments, for instance.