Euler Line
Centroid, circumcentre and orthocentre are collinear.
Key idea
Centroid, circumcentre and orthocentre are collinear.
In depth
The centroid, circumcentre and orthocentre of any triangle are collinear. The centroid lies one third of the way from circumcentre to orthocentre. For an equilateral triangle they coincide.
Worked example
If circumcentre O is at (0, 0) and orthocentre H at (6, 3), the centroid G = (2, 1).
Real-world uses
- Explaining special alignments of triangle centres in advanced geometry and design of symmetric shapes
Common mistake
Applying the claim to the incentre, which generally is not on the line.
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 Euler Line?
Centroid, circumcentre and orthocentre are collinear. It is a topic at the expert level in Geometry.
Where is Euler Line used in real life?
Real-world examples: Explaining special alignments of triangle centres in advanced geometry and design of symmetric shapes.
What should I learn before Euler Line?
Make sure you are comfortable with Orthocentre, the previous topic in Geometry.
What is a common mistake with Euler Line?
Applying the claim to the incentre, which generally is not on the line.
How can I practise Euler Line?
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 Euler Line?
Move on to Nine-Point Circle 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.