I’m a teacher in mathematics at a high school. Math has fascinated me for almost my entire life, so I would like to bring that enthusiasm to my students with beautiful yet easy to understand proofs or demonstrations. It’s meant for students who are in their last grade of high school and will be going to university next year.
So what are simple proofs or visual examples that made you love math? The more examples the better! Answers with pictures would be even better!
Thanks in advance!
P.S. Things that I did already teach my students the basics of are: complex numbers, probability theorem, prime numbers, vectors, functions of more variables, a little bit about group theory, set theory. These are all things that I tried to mix with the things they should actually know for their exams. It’s meant to give them an idea of what math is really about, not just repeating formulas.
Here is one example that I find aesthetically pleasing, and which I have
found effective in 8th-grade classrooms.
Suppose you desire to cut out a triangle from the middle of a piece of paper,
not by punching the scissors through and cutting
the perimeter, but rather by folding the paper
and then cutting
straight through the folded paper.
The natural solution is to mountain-crease (red below) the angle bisectors,
and valley-crease (green dashed) a “perpendicular” from the incenter $x$:
What I find so pleasing is that when you perform this physically, the
angle bisectors meet at a point $x$ (the incenter), and one grasps
Proposition 4, Book IV of Euclid viscerally. Naively, it could well
be that the bisectors do not meet at a point. But careful creasing
shows experimentally that they do.
I have found this tactile demonstration more convincing to
(U.S.) 8th-graders than a two-column Euclidean proof.
(This repeats content from my answer to a related question.)