My first example shows a very standard way to construct a parabola by using a focus point and a directrix line. Not least does this method give understanding of the reflection properties of the parabola, that is hard to show in "classical" ways. (Click the Back-button on your web browser when you want to return to this side.)
There are simple ways to construct all conics as envelopes of lines.
This last example gave me the idea to look for something corresponding to the directrix line of the parabola. What I found was directrix circles for the other conics. A little bit to my surprise the very same construction that gives an ellipse, also gives a hyperbola. Also here the reflection properties come out very nicely.
A rectangle is inscribed in an acute angled triangle. This sentence turns out to open up a problem with an enormous richness.