Nine-Point Circle
A circle through nine special points of a triangle.
Key idea
A circle through nine special points of a triangle.
In depth
For any triangle, nine special points lie on one circle: midpoints of the sides, feet of the altitudes, and midpoints of segments from the orthocentre to the vertices. Its radius is half the circumradius.
Worked example
If the circumradius is 10, the nine-point circle has radius 5.
Real-world uses
- Advanced geometry in olympiads
- Understanding how special triangle points relate
Common mistake
Forgetting that its centre is the midpoint of the circumcentre and orthocentre.
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 Nine-Point Circle?
A circle through nine special points of a triangle. It is a topic at the expert level in Geometry.
Where is Nine-Point Circle used in real life?
Real-world examples: Advanced geometry in olympiads; Understanding how special triangle points relate.
What should I learn before Nine-Point Circle?
Make sure you are comfortable with Euler Line, the previous topic in Geometry.
What is a common mistake with Nine-Point Circle?
Forgetting that its centre is the midpoint of the circumcentre and orthocentre.
How can I practise Nine-Point Circle?
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 Nine-Point Circle?
Move on to Ceva’s Theorem next.
Related expert topics in Geometry
- Power of a Point – For a point and a circle, products of segment lengths along secants are equal.
- 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.