In my geometry class, when I get to the section on triangle “centers”, there are three main ones that have good mathematical and practical utility that make for wonderful application: the circumcenter (center of a circle containing all three vertices), incenter (center of largest inscribed circle), and the centroid (actual balance point of any triangle). But when I get to a fourth center - the *orthocenter* (which is the concurrent intersection of all three extended altitude lines of any triangle) I have thus far been stymied to offer up any use, application, or even mathematical utility for this particular center. [And yes - I know about the *Euler Line* which includes the orthocenter in its interesting set - but I only know that as a wonderful mathematical factoid to have fun with more than an application toward anything further.]

I usually take the occasion to tease students with the ongoing challenge mathematicians aspire to, to find something so totally cool and yet so challengingly useless that nobody is able to apply or use it toward anything else - and how mathematicians have been foiled at pretty much every turn because somebody will end up brilliantly using their hopeful conjecture or new theorem as a springboard toward something practical or some further mathematical utility.

So I also have to include the obligatory humility that I’m pretty sure the orthocenter probably does have a use (multiple ones I shouldn’t wonder) and that I’m just as-of-yet ignorant of them. Not that I’ve spent any large portion of my hours officially researching this - but simple internet searches haven’t turned up anything satisfactory for me. And some quick results just yield up answers from people who are mistaken about what they’re talking about (pretty sure the person answering this question was confusing ‘orthocenter’ with ‘circumcenter’ - the more proper answer to the question posed. …though the glider answer given further down might be intriguing).

As you can see, the orthocenter is one of those ‘centers’ that doesn’t even have to remain in the interior of the actual triangle, as seen in the obtuse triangle example shown below. Does anybody out there happen to know of some architectural or engineering significance - or even further mathematical utility that the orthocenter of a triangle manages to fulfill? I offer that up here as a fun amusement to ponder.