The nine-point circle is a circle that can be that can be constructed for any given triangle. It is named the nine-point circle as it passes through nine points defined from the triangle:
- The midpoint of each side of the triangle (Ma, Mb, Mc);
- The foot of each altitude (Ha, Hb, Hc);
- The midpoint of the line segment from each vertex of the triangle to point where the three altitudes meet, i.e. the orthocentre H, (Ea, Eb, Ec).
Note that for an acute triangle, six of the points – the midpoints and altitude feet – lie on the triangle itself. On the contrary, for an obtuse triangle two of the altitudes have feet outside the triangle, but these feet still belong to the nine-point triangle.
The nine-point circle is the complement to the circumcircle, which is the unique circle that passes through each of the triangle’s three vertices.
Although credited for its discovery, Karl Wilhelm Feuerbach did not entirely discover the nine-point circle, but rather the six point circle, as he only recognised that the midpoints of the three sides of the triangle and the feet of the altitudes of that triangle lay on the circle. It was mathematician Olry Terquem who was the first to recognise the added significance of the three midpoints between the triangle’s vertices and the orthocenter.
Three Properties of the Nine-Point Triangle
- The radius of a triangle’s circumcircle is twice the radius of the same triangle’s nine-point circle.
- A nine-point circle bisects a line segment going from the corresponding triangle’s orthocenter to any point on its circumcircle.
- All triangles inscribed in a given circle and having the same orthocenter have the same nine-point circle.
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