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Japanese Sacred Geometry

Problem 1154: The Sangaku Three-Circle and Tangent Line Theorem

Discover the elegant relationship between the radii of three mutually tangent circles on a straight line.
Problem Statement
In the figure below, three circles with radii $a$, $b$, and $c$ are mutually tangent to each other and to a common straight line. Prove that:
$$\frac{1}{\sqrt{c}} = \frac{1}{\sqrt{a}} + \frac{1}{\sqrt{b}}$$
Geometry problem 1154 Sangaku Problem: Three circles and a tangent line.
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👉 Explore the Original Mathematical Version – Dive into the core proof and foundation.
🎨 View Geometric Art Version 1 – See this beautiful theorem visualized as sacred geometry.