A relation associated with a fixed circle (or conic) and an external or internal point: the power of a point P with respect to a circle with center O and radius r equals PO^2−r^2 and equals the product of signed distances from P to the two intersection points of any line through P with the circle.

Principle

Metric incidence: for any line through P meeting the circle at X and Y, the directed product PX·PY is constant and equals the algebraic quantity PO^2−r^2; this characterizes how P sits relative to the circle (inside, on, outside).

Demonstration

Given circle (O,r) and point P outside it, draw a secant through P meeting the circle at X and Y; compute PX·PY and check it equals PO^2−r^2. If P is on the circle the power is zero; if inside the product uses signed segments and is negative of squared distance difference.

Misapplication

Using unsigned lengths without regard to sign and directed segments can miscompute the product, or applying the formula to non-conic curves where the product-of-segments property fails; treating intersection multiplicities incorrectly at tangency leads to mistakes.

Consequence

Gives a unifying algebraic invariant for many circle configurations: detects tangency (power zero), provides chord length relations, supports radical-axis constructions and coaxal system reasoning, and reduces certain locus problems to simple algebraic equations.

Reversal

Negating the viewpoint yields the radical-axis perspective: instead of a single point's power to one circle, consider equal powers to two circles which locate a line (their radical axis). This inverts point-based invariants into line loci of equal power.

Boundary

Applies to circles (and by extension conics under appropriate projective or metric interpretations) but not to arbitrary curves; requires a well-defined notion of signed distances or an algebraic embedding to interpret PO^2−r^2 over the ground field.

Semantic Tension

Often conflated with purely Euclidean distance statements; the tension is between treating the power as a metric squared-distance quantity versus as an algebraic invariant used in projective or inversion-based methods.

Synthesis

The power of a point is the constant signed product of directed segments from the point to a circle's intersection points with any line through it, equal algebraically to PO^2−r^2 and central to radical-axis and tangency arguments.