A follow-up/corollary to our previous post: yet another characterization of the triangle whose nine-point center coincides with one of its vertices, namely the
-triangle. pdf here.

In
, let
and
. PROVE that the nine-point circle of
coincides with the incircle of
, where
is the orthocenter.






See the pdf here.
Suppose that the nine-point circle of
coincides with the incircle of
, where
is the orthocenter. PROVE that
and
.





Same link as above: pdf here.
Takeaway
In any triangle , let
be the side-lengths,
the orthocenter,
the circumcenter, and
the circumradius. Then the following statements are equivalent:
is equilateral
- the reflection of
over
is
- the nine-point circle of
is the same as the incircle of
- the reflection of
over
is
and the reflection of
over
is
Task
- (Aufbau) In triangle
, let
be the side-lengths,
the circumradius,
the circumcenter,
the nine-point center, and
the orthocenter. PROVE that the following statements are equivalent:
is equilateral
- the reflection of
over
is
- the area
satisfies
- the circle with diameter
passes through vertices
and
- the nine-point circle of
is the same as the incircle of
- radius
is parallel to side
and radius
is parallel to side
- the reflection of
over
is
and the reflection of
over
is
is the geometric mean of
and
, and
is the geometric mean of
and
. (
and
are the feet of the altitudes from
and
, respectively.)