Of the nine main points through which the nine-point circle passes, three of them are Euler points: the midpoints of the line segments joining the orthocenter to each vertex (for example, the three green dots in the diagram below).

(1)

only eight (rather than nine) points are present in the above nine-point circle, as the Euler point is forced to coincide with the foot of the altitude from vertex . At the same time, the two Euler points and correspond to the reflections of the midpoints of sides and over side .

First suppose that and let be the radius of the circumcircle of triangle .

By definition of the Euler points, is the midpoint of the segment joining vertex to the orthocenter . Since in any triangle, we have that . By the assumption we then have:

By one of the equivalent statements here, we conclude that equation (1) holds. The converse is similar.

Just as in example 1 above.

Consider . Since from example 1 and by definition, we have that is isosceles with base . The altitude through bisects the base and so is a reflection of over side .

Consider and use example 2.

Just as in example 1.

## Takeaway

In triangle , let denote the Euler points, the foot of the altitude from vertex , and the midpoints of sides . Then the following statements are *equivalent*:

- is the reflection of over side
- is the reflection of over side
- the side-lengths satisfy equation (1)
- the chord is a diameter of the nine-point circle

More in the task below.

## Task

- (Early 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-two*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
- 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 .

( short of the target.)

- (Extra feature) If satisfies equation (??), PROVE that its nine-point center divides in the ratio .