Another characterization of the triangle whose nine-point center coincides with one of its vertices, namely
-triangle, is discussed here.

In
, let
and
. PROVE that
is equilateral, where
is the orthocenter.





See the pdf.
In
, suppose that
is equilateral, where
is the orthocenter. PROVE that
and
.





Same link as above: pdf.
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 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
- 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.)