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# Euler points as reflected midpoints

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).

Owing to the equation

(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 .

In , let be the Euler point associated with vertex . PROVE that equation (1) implies (and is implied by) .

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.

In , let be the Euler point associated with vertex . PROVE that equation (1) implies (and is implied by) .

Just as in example 1 above.

If the side-lengths of satisfy equation (1), PROVE that is the reflection of over side .

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 .

If the side-lengths of satisfy equation (1), PROVE that is the reflection of over side .

Consider and use example 2.

In , let be the Euler point associated with vertex . PROVE that equation (1) implies (and is implied by) .

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:

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

• (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:
1. or
2. is the reflection of over side
3. is the reflection of over side
4. is congruent to
5. is congruent to
6. is isosceles with
7. is isosceles with
8. is right angled at
9. is the circumcenter of
10. is right-angled at
11. is right-angled at
13. the points are concyclic with as diameter
14. the reflection of over lies internally on
15. the reflection of over lies externally on
16. radius is parallel to side
17. is the reflection of over side
18. the nine-point center lies on
19. the orthic triangle is isosceles with
20. the geometric mean theorem holds
21. the bisector of has length , where
22. the orthocenter is a reflection of vertex over side
23. segment is tangent to the circumcircle at point
24. median has the same length as the segment
25. the bisector of is tangent to the nine-point circle at
26. is a convex kite with diagonals and
27. altitude is tangent to the nine-point circle at
28. chord is a diameter of the nine-point circle
29. 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 .