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.)