ALTITUDES and ORTHOCENTERS of triagles reveal some interesting observations. CLICK HERE to see two different ways to construct a NINE POINT CIRCLE. If you have access to GSP, you may want to CLICK HERE to see a version that you can play around with.
CLICK HERE to see my efforts at constructing a triangle of minimal perimeter inside a given triangle. You may want to try it yourself before looking at my work. You can also see the same using GSP by clicking here.