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The nine-point circle is also called Euler's circle. It passes through the intersection of the heights of a triangle with its corresponding sides (also called perpendicular feet). So these are the first three points through which the circle passes. In 1765 Euler showed that this circle also passes through the midpoints of the sides of a triangle. In 1822 Feuerbach showed that the nine-point circle passes also through the tree midpoints of the segments that join the vertices with the orthocenter of the circle. Hence the circle is also sometimes called Feurbach circle (Euler would have probably called it six-point one!).

The radius of the nine-point circle is exactly half of the radius of the circumcircle.

Let us examine this circle in more detail. First, it contains the three points which are 'perpendicular feet' of the triangle ABC - these are the points where the heights of the triangle ABC which are dropped from the vertices, intersect with the opposite sides. These points have been labelled as E, F, and G.

The second set of three points is the set of the points which are midpoints of the sides of the triangle ABC. These points are now labelled as H, I, and J.

Euler apparently knew about these six points described so far: E, F, G, H, I, and J.

Feuerbach in 1822 discovered that the nine-point circle also passes through the three points which are midpoints of the segments that join the verticies with the orthocenter of the circle. Click here to see the explanation of what the orthocenter is.

These points have now been labelled as K, L, and M.

• If you draw a circumcircle (green in the drawing, click here to find more about the circumcircle), then any line which joins any point on it, with the orthocentre of a triangle, will be divided exactly in half by the ninepoint circle. In the drawing below, the orthocentre is labelled D, and the line from it to the circumcircle is labelled DN, having a midpoint at O.

• All triangles incscribed in a circle (so that would be their circumcircle), and having the same orthocentre, will also share the same nine-point circle.

or about various properties of triangles:

Euler Line

Circumcentre

Centroid

Orthocentre

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