• thank you, sal.ï»¿ however, i remember, while in school, i used to always think "yeah, but...."(meaning, what are all of these things used for in the real world?). i know your main concern is getting the information across, but it's never clear how these things get implemented into the real world. is it helpful in programming, or software engineering, or architecture, game design...? probably, but how?

• The power of mathematics is that it can abstract problems from many different domains AND tell us fundamental things about our universe (which may also turn out to be useful). It really is a purer philosophy (not jaded by fuzzy word play) that also turns out to be the coreï»¿ of science and technology. When I first learned about Euler's Line, it gave me goosebumps. Tells me that there is a deep structure to the universe that we are only beginning to catch glimpses of.

• I find it sad that folks see mathematics as a tool to be used in "the real world". That's one aspect of it, but give math some credit as a field of its own. I'd like to see someone list the applications of engineering in math for a change. Mathematics is more than just a tool. It's a pursuitï»¿ in its own right. I mean, we don't teach performing arts just so people can compete in today's dramatic world; we have performing arts class because performing arts is beautiful.

• This looks very interesting but I'm not old enough to know half the vocabulary Sal saysï»¿ :[

sir, can we prove that using co-ordinate geometry too?ï»¿

• And prove thatï»¿ it is parallel to LOL!

• Do bubbles observe these trigometric laws of Euler's triangles in 3D? I would think so - which would be quite interesting. Resonance nodes andï»¿ locations of peaks would also be identified...if the surfaces or wires (strings) were to be uniform along the axes you are defining. Cool extrapolations can be made.

• 2 sides proportional and 1 congruent angle for each triangle = Similar triangles. I remember that. Amazing how Euler used thatï»¿ for his proof.

• This sort of thing is indeedï»¿ extremely useful in game programming. Geometry is at the core of collision detection algorithms. Still, like Khan was saying, mathematics has a beauty to it that can be appreciated outside of practical application. One need only foster a desire for truth to see it. For many people, application is all that matters. Everyone has their own perspective, and neither is right or wrong. But there is something transcendental about loving math for its own sake.

• Just AWESOME. The next time I have some free time (I'm trying to schedule for sometime in 2015) I'm going to get some good material on this stuff and plunge into it. It really PROFOUNDLY FASCINATES me....

Thanks for bringing this moving information to us, Sal.ï»¿

• in your last video, you said that this line has a magical relationship with Euler's formula. Can you show me what is it Sal? plzï»¿ plz plz