Answer to Question #227 in Geometry for Otto James

Question

The distance between the centers of circles R1 and R2 is d. Prove that these circles are orthogonal only if d² = R1² + R2².

Answer 1

Let the circles with centers O1 and O2 pass through the point A. Radiuses are perpendicular to the circle tangents at the point A, thus the circles are orthogonal only if the angle ∠O1AO2 = 90°, that's why O1O2² = O1A² + O2A².