nine-point circle

The nine point circle also known as the Euler’s circle or the Feuerbach circle is the circle that passes through the feet of perpendiculars from the vertices $A, B$ and $C$ of a triangle $\\triangle ABC.$

Some of the properties of this circle are:

Property 1 : This circle also passes through the midpoints of the sides $A B, B C$ and $C A$ of $\\triangle ABC.$ This was shown by Euler.

Property 2 : Feuerbach showed that this circle also passes through the midpoints of the line segments $A H, B H$ and $C H$ which are drawn from the vertices of $\\triangle ABC$ to its orthocenter $H .$

These three triples of points make nine in all, giving the circle its name.

Property 3 : The radius of the nine-point cirlce is $R/2,$ where $R$ is the circumradius (radius of the circumcircle).

Property 4 : The center of the nine-point circle is the midpoint of the line segment joining the orthocenter and the circumcenter, and hence lies on the Euler line.

Property 5 : All triangles inscribed in a given circle and having the same orthocenter, have the same nine-point circle.

单词	NinepointCircle
释义	nine-point circle The nine point circle also known as the Euler’s circle or the Feuerbach circle is the circle that passes through the feet of perpendiculars from the vertices $A, B$ and $C$ of a triangle $\\triangle ABC.$ Some of the properties of this circle are: Property 1 : This circle also passes through the midpoints of the sides $A B, B C$ and $C A$ of $\\triangle ABC.$ This was shown by Euler. Property 2 : Feuerbach showed that this circle also passes through the midpoints of the line segments $A H, B H$ and $C H$ which are drawn from the vertices of $\\triangle ABC$ to its orthocenter $H .$ These three triples of points make nine in all, giving the circle its name. Property 3 : The radius of the nine-point cirlce is $R/2,$ where $R$ is the circumradius (radius of the circumcircle). Property 4 : The center of the nine-point circle is the midpoint of the line segment joining the orthocenter and the circumcenter, and hence lies on the Euler line. Property 5 : All triangles inscribed in a given circle and having the same orthocenter, have the same nine-point circle.
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