You are currently browsing the category archive for the ‘geometry’ category.

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:

Screen Shot 2017-09-30 at 4.22.58 PM

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

Screen Shot 2017-09-30 at 4.23.12 PM


Step 3: Draw the circle through the point D.

Screen Shot 2017-09-30 at 4.23.19 PM

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).

Screen Shot 2017-09-30 at 5.02.59 PM


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.

Screen Shot 2017-09-30 at 5.07.40 PM.png


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).

Screen Shot 2017-09-30 at 5.10.41 PM

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

Screen Shot 2017-09-30 at 5.14.51 PM

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

Screen Shot 2017-09-30 at 5.30.19 PM

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.

Screen Shot 2017-09-30 at 5.33.33 PM

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)

Screen Shot 2017-09-30 at 5.36.49 PM


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.

Screen Shot 2017-09-30 at 5.45.06 PM

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

Screen Shot 2017-09-30 at 6.08.06 PM

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

Screen Shot 2017-09-30 at 6.08.37 PM

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

Screen Shot 2017-09-30 at 6.08.26 PM

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

Screen Shot 2017-09-30 at 6.16.38 PM

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

Website Stats


Recent Posts

Follow IB Maths Resources from British International School Phuket on