Have you ever wondered what mathematicians mean when they talk about a “beautiful” math proof?
“Beauty in mathematics is seeing the truth without effort.”
“There’s something striking about the economy of the counselor’s construction. He drew a single line, and that totally changed one’s vision of the geometry involved.
“Very often, there’s a simple introduction of something that’s not logically within the framework of the question — and it can be very simple — and it utterly changes your view of what the question really is about.”
CREDITS: Castle photo (top) by Rachel Davis via Unsplash. “A Mathematical Fable” via YouTube. Story told by Barry Mazur. Animation by Pete McPartlan. Video by Brady Haran for Numberphile.
Denise Gaskins' Let's Play Math 
Wonderful video, thank you for sharing this. If I had access to videos like this in high school (or earlier), math would have been so much fun, rather than drudgery with little point.
I always wondered in geometry class WHY we should care about the angles of a triangle. This video shows a practical use in terms kids can understand. There is also a great children’s book available – What’s Your Angle Pythagoras? that tells a story of a young Pythagoras using his famous theory to help his father: https://www.charlesbridge.com/products/whats-your-angle-pythagoras
I believe some textbooks now teach geometry beginning with symmetry, similarity, and transformations. It’s a beautiful approach, and more intuitive for many students.