Nine point circle
The nine point circle (also known as Euler's circle or Feuerbach's circle) of a given triangle is a circle which passes through 9 "significant" points:
- The three feet of the altitudes of the triangle.
- The three midpoints of the edges of the triangle.
- The three midpoints of the segments joining the vertices of the triangle to its orthocenter. (These points are sometimes known as the Euler points of the triangle.)
That such a circle exists is a non-trivial theorem of Euclidean geometry.
The center of the nine point circle is the nine-point center and is usually denoted .
Proof of the Existence
Since is the midpoint of
and
is the midpoint of
,
is parallel to
. Using similar logic, we see that
is also parallel to
. Since
is the midpoint of
and
is the midpoint of
,
is parallel to
, which is perpendicular to
. Similar logic gives us that
is perpendicular to
as well. Therefore
is a rectangle, which is a cyclic figure. The diagonals
and
are diagonals of the circumcircle. Similar logic to the above gives us that
is a rectangle with a common diagonal to
. Therefore the circumcircles of the two rectangles are identical. We can also gain that rectangle
is also on the circle.
We now have a circle with the points ,
,
,
,
, and
on it, with diameters
,
, and
. We now note that
. Therefore
,
, ad
are also on the circle. We now have a circle with the midpoints of the sides on it, the three midpoints of the segments joining the vertices of the triangle to its orthocenter on it, and the three feet of the altitudes of the triangle on it. Therefore the nine points are on the circle, and the nine-point circle exists.
We know that the reflection of the orthocenter about the Triangle's sides and about the mid points of the triangle's sides lie on the circumcircle.
Thus consider the homothety centred at
with ratio
.It maps the circumcircle to the nine point circle.
Hence proved.
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