1847 Sangaku problem

Sangaku were geometrical problems, often presented in a colourful diagram on wooden tablets as challenges for interested people to work on

they were placed by people from all walks of life in the vicinity of Shinto shrines and Buddhist temples during the Edo period (1603 - 1867)

during this time Japan was mostly isolated from the influences of other countries


this problem is claimed to have been posed by a 13 year old person

a square is inscribed within a right angled triangle

three blue circles have the same radii and are inscribed as shown

the pink circle is inscribed in the lower right angled triangle

what are the radii of the blue circles and pink circle in terms of the radius of the orange circle?


it needs some reasonable hefty algebraic manipulation using Pythagoras' rule (twice): within the square (using where the circles meet) and in the upper right angled triangle

you should find the 3 , 4 , 5 triangle emerges from this

then a simple application of similarity from the top right angled triangle to the lower triangle can be used to establish that the pink radius is twice the orange radius