I've been grappling with the following puzzle for the past few days, off and on.

The outer square has sides of length a. Find the side length of the internal square, as well as the radii of the two internal circles in terms of a.


50 Adam Points for anyone who gives me all three correct answers!

From this nice puzzle sit (on the medium page):

Note: This geometry problem comes from the tradition of sangaku. During the period between 1639 and 1854, many Japanese samurai apparently adopted geometry as a serious mental discipline! They would solve many difficult geometric problems, and carefully inscribe the solutions into beautiful wooden tablets, often with color. Then they would hang their solutions under roofs for public viewing, either to show respect for the elegance of math, or perhaps just to show off their intelligence and challenge other practitioners of sangaku -- a Japanese word that literally means mathematical tablet. Sangaku problems are usually just Euclidean geometry, but others seem almost impossible to do without cheating and using higher level math (e.g. calculus, affine transformations). Also notable is a strong emphasis on ellipses and circles, an emphasis not found in Western studies of geometry.

Note 2: For a hardcopy of the above image only, click here and print. Hardcopies are useful for drawing lines and labeling vertices and what not.

Problem source: Fukagawa, H. and D. Pedoe. Japanese Temple Geometry problems. Charles Babbage Research Foundation: Winnipeg, 1989.