##  [Power of a Point](/power-point-0) 

 Definition

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

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

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

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

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

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

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

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

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.