Nine Point Circle 

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The ninepoint 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 ninepoint 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 sixpoint one!). The radius of the ninepoint 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 ninepoint 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. Some really really interesting things about this circle are the following facts:
To see how you can construct a ninepoint circle click here.

Learn more about Circle here or about various properties of triangles: Learn more about Euler or some other famous mathematicians. Some interesting mathematical artefacts can be found here! Mathematicians' delights and possessions... 

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