Among all triangles, the only triangle whose nine-point center coincides with one of its vertices is the -triangle.
As such, it has unique properties, some of which will be shown in today’s post.
Additionally, the -triangle is also unique among all isosceles triangles, and we’ll see that later.

Let be the circumradius and let
be the side-lengths of a given triangle
. By the extended law of sines we have:
In the event that , we get
.

Suppose that vertex of triangle
is equidistant from the midpoints of sides
. In particular, this means that the median from vertex
is equal to half the side-lengths of
and
.
By the cosine formula:
Consequently, .

Note that the side-lengths are in the ratio
, and so
. Consider the length of the median from vertex
:
And so vertex is equidistant from the midpoints of all three sides.

Let the feet of the altitudes be , and let the altitudes themselves be
. Suppose that vertex
is equidistant from
and
, that is
. Then:
Similarly, let :
And so . Since
is equivalent to
, we obtain
. In turn,
. The fact that
then gives
.

Easy converse to the preceding example.
Takeaway
In any triangle , let
be the side-lengths and
the circumradius. Then the following statements are equivalent:
is equidistant from the midpoints of the sides
is equidistant from the feet of the three altitudes
is equidistant from the three Euler points.
Such must be the nine-point center.
Task
- (Late seventies) In a non-right triangle
, let
be the side-lengths,
the altitudes,
the feet of the altitudes from the respective vertices,
the midpoints of sides
,
the Euler points,
the circumradius,
the circumcenter,
the nine-point center,
the orthocenter,
the reflection of
over side
,
the reflection of
over side
, and
the reflection of
over side
. PROVE that the following seventy-eight statements are equivalent:
or
is the reflection of
over side
is the reflection of
over side
is congruent to
is congruent to
is isosceles with
is isosceles with
is right angled at
is the circumcenter of
is right-angled at
is right-angled at
- quadrilateral
is a rectangle
- the points
are concyclic with
as diameter
- the reflection
of
over
lies internally on
- the reflection
of
over
lies externally on
- radius
is parallel to side
is the reflection of
over side
- segment
is perpendicular to side
- the nine-point center lies on
- the orthic triangle is isosceles with
- the geometric mean theorem holds
- the bisector of
has length
, where
- the orthocenter is a reflection of vertex
over side
- segment
is tangent to the circumcircle at point
- median
has the same length as the segment
- the bisector
of
is tangent to the nine-point circle at
is a convex kite with diagonals
and
- altitude
is tangent to the nine-point circle at
- chord
is a diameter of the nine-point circle
- segment
is tangent to the nine-point circle at
.
(Almost losing count with more still lined up.)