Sangaku Math Fixed < 720p 2025 >

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)).

From first equation: [ (h - R)^2 + (x - R)^2 = (R + x)^2 ] [ (h - R)^2 + x^2 - 2Rx + R^2 = R^2 + 2Rx + x^2 ] [ (h - R)^2 - 2Rx = 2Rx ] [ (h - R)^2 = 4Rx ] [ h - R = 2\sqrt{Rx} \quad \Rightarrow \quad h = R + 2\sqrt{Rx} ] sangaku math

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). Place the line as the x-axis: (y=0)

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} ] From first equation: [ (h - R)^2 +

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 ]