I had been reading about orthocenter properties on the web one day when I thought that you might be able to show some of its properties using a tactile activity. I tried it out during my prep period, and it worked as I imagined! Pretty neat.
Here are some pictures, taking you through the steps.
Step 1: Draw a circle.
Step 2: Draw any triangle inscribed inside the circle. (Now is a good time to introduce vocabulary words like "inscribed" and "circumscribed"...)
Step 3: Cut out the circle, and fold it inwards along the edges of the triangle.
Step 4: Mark the point where the three arcs coincide. This is the orthocenter!
Step 5: Verify that your orthocenter is indeed the intersection of the three altitudes by connecting each vertex with the orthocenter, and making sure the result looks perpendicular to the opposite edge.
Step 6: To explain why it works, we label the sides that are congruent with tickmarks!
It doesn't help make orthocenters useful, but it is a fun/easy tactile activity (very 9th-grade appropriate, methinks) that shows visually some of the deeper mathematical properties about orthocenters.
Addendum: Oops - sorry, I was being sloppy with the vocabulary earlier. Can you tell I have circumcenter on my mind? :)