Tanglänge

Tanglänge is a geometric term used to denote the length of a tangent segment drawn from a point to a circle. It is a basic concept in circle geometry and relates to the power of a point.

Definition and formula: Consider a circle with center O and radius r. Let P be a point outside the circle, and let PT be the tangent segment from P to the circle, touching it at T. The Tanglänge is the length PT. If d denotes the distance from P to the center O (d = OP), then PT^2 = d^2 − r^2. Therefore PT = sqrt(OP^2 − r^2). If P lies on the circle, PT = 0; if P is inside the circle, a real tangent segment from P to the circle does not exist in the usual sense.

Coordinate form and power of a point: For a circle centered at the origin with radius r, and a point P at distance d from the origin, PT = sqrt(d^2 − r^2). More generally, PT^2 equals the power of P with respect to the circle. This connects Tanglänge to other circle theorems.

Properties: From a given external point P, the two possible tangent segments to a circle have equal length, so PT1 = PT2. The Tanglänge can be interpreted as the square root of the point’s power with respect to the circle. If a line through P meets the circle at A and B, then PA × PB equals PT^2 (the tangent power).

Applications: The concept is used to solve distance problems in Euclidean geometry, analyze tangent lines, and relate tangent lengths to secants and the power of a point.