Olympiad Right Triangle Inequality (Canada 1969 - visual proof)

This is a short, animated proof without words demonstrating an inequality about the side lengths of a right triangle that was Problem 3 in the 1969 Canadian Mathematical Olympiad. #manim #math #mathvideo #righttriangle #inequality #mathshorts #geometry #animation #theorem #pww #proofwithoutwords #visualproof #proof #algebra​​ ​ #proof #mathematics​​ #mtbos

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This animation is based on a visual proof from Roger B. Nelsen's second proofs without words compendium (page 12).

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Music in this video:
Sprite Star from Saidbysed

[_Heb_]

My solution wasn't quite as nice, but I found if you square both sides of the inequality you get a^2 + b^2 + 2ab <= 2c^2, which simplifies to 2ab <= c^2. Then, if you take four of the original triangle and arrange them into a square with each hypotenuse as a side length, you get a square (with a hole in the middle) of area c^2. Since the four triangles (combined area 4*ab/2 = 2ab) fit entirely within the large square, it must be that the triangles' area <= the square's area

[shaharjoselevich]

Woah! It's pretty obvious using the Cauchy-Schwarz inequality and Pythagoras, but this way is much more visual and beautiful!
Also nice animation

[wangryanlihci]

Hello, I was wondering what application you used to make these proofs. I am currently working on a mathematical project, and explaining it to the judges with an animated proof would be so much more useful than explaining it with words. Btw this is slick, really cool.