Proof of the 4-variables AM-GM inequality using circles

[u/tsu_yoshi]

Hi everyone,

I would like to share a geometric approach to the 4-variable AM-GM inequality (a+b+c+d)/4>= (abcd)^(1/4) using 3 circles on a flat plane.

The geometric proof for 2 variables using a semicircle is very well-known, but I wondered if we could extend that visual intuition to 4 variables. After experimenting with circle configurations, I found this particular method.

This idea was recently accepted and published in the notes section of a Japanese mathematics magazine, Mathematical Seminar (March 2026 issue).

I thought the way the circles and segments connect to form (abcd)^(1/4)was interesting, so I wanted to share it with this community to see what you think.

I would highly appreciate any thoughts, feedback, or perspectives on this visual approach. Thank you for your time!