Power of a Point
For a point and a circle, products of segment lengths along secants are equal.
Key idea
For a point and a circle, products of segment lengths along secants are equal.
In depth
For a point P and a circle, any line through P meeting the circle at A and B gives the same product PA × PB. For a tangent it equals PT². It follows from similar triangles.
Worked example
Two chords cross at P with pieces 3 and 8 on one, and 4 and x on the other: 3 × 8 = 4x → x = 6.
Formula
Real-world uses
- Solving puzzles about intersecting circles and lines
- Advanced geometry in competition maths
Common mistake
Using the whole chord instead of the segments from P.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Power of a Point?
For a point and a circle, products of segment lengths along secants are equal. It is a topic at the expert level in Geometry.
What is the formula for Power of a Point?
The key formula is PA·PB = PC·PD.
Where is Power of a Point used in real life?
Real-world examples: Solving puzzles about intersecting circles and lines; Advanced geometry in competition maths.
What should I learn before Power of a Point?
Make sure you are comfortable with Chord and Tangent Properties, the previous topic in Geometry.
What is a common mistake with Power of a Point?
Using the whole chord instead of the segments from P.
How can I practise Power of a Point?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Power of a Point?
Move on to Cyclic Quadrilaterals next.
Related expert topics in Geometry
- Cyclic Quadrilaterals – Opposite angles of a quadrilateral inscribed in a circle sum to 180°.
- Cavalieri’s Principle – Solids with equal cross-sectional areas at every height have equal volume.
- Area by Coordinates (Shoelace) – Area of a polygon from vertices.
- Parabola (Geometric Definition) – Points equidistant from a focus and a directrix.
- Ellipse – Sum of distances to two foci is constant.
- Hyperbola – Difference of distances to two foci is constant.