The nine-point circle

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Sometimes it’s worth remembering something simply because it’s neat. In this case it’s called the ‘nine-point circle’. Sometimes called Euler’s circle, sometimes Feuerbach’s circle (there are numerous other variations), this is a classic example of how a little exploration with a ruler and compass can often turn up completely unexpected relationships. And if you care to delve deeper on the internet you’ll discover that the idea has been taken much further, turning up even more obscure and surprising qualities.

For this design we’ve taken the simplicity of the basic nine-point circle. It’s constructed as follows:

  • Take a triangle (in this case we’ll use one with only acute angles) and draw lines from each angle to the middle of the opposite side, like this:

Explainer-from-Geogebra-01 We’ve labelled each corner A, the points where the perpendiculars meet the sides as B and point where they meet – called the ‘orthocentre’ of the triangle C. We have no need to label every point with a different letter since we’ll be doing the same thing to each line.

  •  Now we mark half-way between point A and point C on each line:


Now we mark the half-way point of each side and draw a triangle between them:

Explainer-from-Geogebra-03Mark the middle points of the sides of the new, smaller, triangle and draw perpendicular lines from those points and call the point where they meet E. For any triangle that point of meeting is the centre of the triangle’s ‘circumscribed circle’ – the circle which passes through the three corners. Every triangle (or any three points) has a unique circle.

Explainer-from-Geogebra-04 Draw the circumscribed circle for the small triangle and notice how it passes through all of the 9 of the significant points that we’ve marked – the 9-point circle.

Explainer-from-Geogebra-05If you’d like to explore the idea further and don’t rate your skills with a compass and straight edge, I’d recommend a wonderful free piece of software called Geogebra. You can find it here and it’s a great tool for playing around with geometric ideas, automating all kinds of otherwise tiresome tasks like finding the mid-point of lines.

Where mathematics becomes an object of desire.