Sangaku Math !full! May 2026

Place the line as the x-axis: (y=0). Let circle (R) have center ((R, R)) — it touches the line at ((0,0)). Let circle (r) have center ((d, r)) with (d > 0), touching the line at ((d, 0)).

1. What Are Sangaku? Sangaku (算額, literally "calculation tablet") are colorful wooden tablets depicting geometric problems, often solved and dedicated to Shinto shrines or Buddhist temples in Japan. They were created by people from all walks of life—samurai, farmers, merchants, and professional mathematicians (called wasanka )—from the early 17th to the late 19th century (the Edo period). sangaku math

Distance between centers of (R) and (r) = (R + r) (external tangency): [ \sqrt{(d-R)^2 + (r-R)^2} = R + r ] Simplify: [ (d-R)^2 + (r-R)^2 = (R+r)^2 ] [ (d-R)^2 + R^2 - 2Rr + r^2 = R^2 + 2Rr + r^2 ] [ (d-R)^2 - 2Rr = 2Rr ] [ (d-R)^2 = 4Rr ] [ d - R = 2\sqrt{Rr} \quad (\text{positive since } d > R) ] [ d = R + 2\sqrt{Rr} ] Place the line as the x-axis: (y=0)

Center = ((h, x)), tangent to line at ((h,0)). Tangency to circle (R): distance between centers = (R + x): [ \sqrt{(h - R)^2 + (x - R)^2} = R + x ] Tangency to circle (r): distance between centers = (r + x): [ \sqrt{(h - (R+2\sqrt{Rr}))^2 + (x - r)^2} = r + x ] They were created by people from all walks