Earlier in the year I went through some of my files from my undergraduate mathematics career. Among the things I found were dynamic geometry files created using the interactive geometry software Cinderella. I created constructions of about 26 different triangle centers based on triangle centers found in Clark Kimberling’s Encyclopedia of Triangle Centers. The goal was to better understand how and why such concurrences translate between plane geometry and spherical geometry. Quickly changing between plane and spherical geometry is one thing that Cinderella is good at and one reason I chose to create the sketches using the program — you can quickly switch between planar view and spherical view. Then using the dynamic and interactive nature of dynamic geometry sketches, you can visually test if a concurrence that creates a triangle center on the plane still seems to create a concurrence on the sphere. Of course, even if a concurrence appears to hold on the sphere, one still has to provide a mathematical proof that this is indeed the case.
The first sketch I’m going to share with you is that of the orthocenter of a triangle — formed from the intersection of the three altitudes of a triangle. You can find out more about the orthocenter of a plane triangle by visiting the entry on the orthocenter on MathWorld.
In any case, if you’re interested, please download the Cinderella file for the orthocenter (orthorcenter.cdy). Try exploring the sketch in different geometrical views. Enjoy! 
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