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:

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

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Step 3: Draw the circle through the point D.

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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).

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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.

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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).

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

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

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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.

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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)

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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.

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

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

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

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

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

Essential resources for IB students:

1) Revision Village

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Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams.  I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.

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There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank of graded questions.  What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

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The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers.   These all have worked solutions and allow you to focus on specific topics or start general revision.  This also has some excellent challenging questions for those students aiming for 6s and 7s.

Each course also has a dedicated video tutorial section which provides 5-15 minute tutorial videos on every single syllabus part – handily sorted into topic categories.

2) Exploration Guides and Paper 3 Resources

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I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.