Nine Point Circle
THE NINE-POINT CIRCLE
The nine-point circle and the associated Euler line bring together several attractive properties of circles and triangles.
![[Team]](https://lh5.googleusercontent.com/Api8XTGOOygLYXAKBwZzGleyvvopy_C76s_zGDf_O4zN3Nw7WJIndX3RKruTZ5bgV_u-bM6o6D2xxOvuLMSEDmuf7qTcNwRjiRzs4fpYD3rdtHB2=w1280)
A triangle has several centres. One is known as the orthocentre, and is the point H where all three altitudes meet. An altitude is a line from a vertex which meets the opposite side of the triangle at right angles. The fact that the three altitudes always meet at a common point is not self-evident, put proofs can be found in geometry texts.
Each triangle also has a unique circle, known as its circumcircle, which passes through each of its three vertices. The point O, equidistant from the three vertices, this distance being the radius of the circumcircle, is known as the circumcentre.
Karl Wilhelm von Feuerbach (1800-1834) showed that there is a unique circle which passes through 6 points, namely the three midpoints of sides and the three points where the altitudes meet opposite sides.
This circle is now known as the nine-point circle, as the midpoint of each line joining the orthocentre to a vertex also lies on this circle.
EULER LINE
The Euler line is the line joining the orthocentre (H) to the circumcentre (O), and is marked in blue in the diagram.
It can also be shown that the medians (lines from vertices to mid-points of opposite sides) meet in a common point (G). This point is known as the centroid. It can be shown that G is always on the Euler line, in such a way that OH=3OG.
Another point always on the Euler line is the centre C of the nine-point circle itself. This is always at the midpoint of OH.
There is another well-known centre of a triangle, the incentre. This is the centre of the unique circle inside the triangle which is tangent to each side. This is also the point where the three angle bisectors of the triangle's vertices meet. This centre is not on the Euler line unless the triangle is isosceles.
We conclude by noting that if, furthermore, the triangle is equilateral, there is no Euler line, as all of the centres above, by symmetry, coincide.
Peter Taylor
March 2012