Euler’s 9 Point Circle

If you are a teacher then please also visit my new site: intermathematics.com for over 2000+ pdf pages of resources for teaching IB maths!

Euler’s 9 Point Circle

This is a nice introduction to some of the beautiful constructions of geometry.  This branch of mathematics goes in and out of favour – back in the days of Euclid, constructions using lines and circles were a cornerstone of mathematical proof, interest was later revived in the 1800s through Poncelot’s projective geometry – later leading to the new field of non Euclidean geometry.  It’s once again somewhat out of fashion – but more accessible than ever due to programs like Geogebra (on which the below diagrams were plotted).  The 9 point circle (or at least the 6 point circle was discovered by the German Karl Wilhelm von Feuerbach in the 1820s.  Unfortunately for Feuerbach it’s often instead called the Euler Circle – after one of the greatest mathematicians of all time, Leonhard Euler.

So, how do you draw Euler’s 9 Point Circle?  It’s a bit involved, so don’t give up!

Step 1: Draw a triangle:

Step 2: Draw the perpendicular bisectors of the 3 sides, and mark the point where they all intersect (D).

Step 3: Draw the circle through the point D.

Step 4: From each line of the triangle, draw the perpendicular line through its third angle.  For example, for the line AC, draw the perpendicular line that goes through both AC and angle B. (The altitudes of the triangle).  Join up the 3 altitudes which will meet at a point (E).

Step 5:  Join up the mid points of each side of the triangle with the remaining angle.  For example, find the mid point of AC and join this point with angle B.  (The median lines of the triangle).  Label the point where the 3 lines meet as F.

Step 6:  Remove all the construction lines.  You can now see we have 3 points in a line.  D is the centre of the circle through the points ABC, E is where the altitudes of the triangle meet (the orthoocentre of ABC) and F is where the median lines meet (the centroid of ABC).

Step 7:  Join up the 3 points – they are collinear (on the same line).

Step 8:  Enlarge the circle through points A B C by a scale factor of -1/2 centered on point F.

Step 9: We now have the 9 point circle.  Look at the points where the inner circle intersects the triangle ABC.  You can see that the points M N O show the points where the feet of the altitudes (from step 4) meet the triangle.

The points P Q R show the points where the perpendicular bisectors of the lines start (i.e the midpoints of the lines AB, AC, BC)

We also have the points S T U on the circle which show the midpoints of the lines between E and the vertices A, B, C.

Step 10:  We can drag the vertices of the triangle and the above relationships will still hold.

In the second case we have both E and D outside the triangle.

In the third case we have E and F at the same point.

In the fourth case we have D and E on opposite sides of the triangle.

So there we go – who says maths isn’t beautiful?

Essential Resources for IB Teachers

1) Intermathematics.com

If you are a teacher then please also visit my new site.  This has been designed specifically for teachers of mathematics at international schools.  The content now includes over 2000 pages of pdf content for the entire SL and HL Analysis syllabus and also the SL Applications syllabus.  Some of the content includes:

1. Original pdf worksheets (with full worked solutions) designed to cover all the syllabus topics.  These make great homework sheets or in class worksheets – and are each designed to last between 40 minutes and 1 hour.
2. Original Paper 3 investigations (with full worked solutions) to develop investigative techniques and support both the exploration and the Paper 3 examination.
3. Over 150 pages of Coursework Guides to introduce students to the essentials behind getting an excellent mark on their exploration coursework.
4. A large number of enrichment activities such as treasure hunts, quizzes, investigations, Desmos explorations, Python coding and more – to engage IB learners in the course.

There is also a lot more.  I think this could save teachers 200+ hours of preparation time in delivering an IB maths course – so it should be well worth exploring!

Essential Resources for both IB teachers and IB students

1) Exploration Guides and Paper 3 Resources

I’ve put together a 168 page Super Exploration Guide to talk students and teachers through all aspects of producing an excellent coursework submission.  Students always make the same mistakes when doing their coursework – get the inside track from an IB moderator!  I have also made Paper 3 packs for HL Analysis and also Applications students to help prepare for their Paper 3 exams.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.