*a circle having as diameter the line segment joining the orthocenter with the centroid*.

Then there’s this modification we want to make. Consider instead a circle having as diameter the line segment connecting the orthocenter with the circumcenter (the Euler line). *This circle passes through vertex of a parent triangle if, and only if, the side-lengths satisfy*

(1)

The reflection of the circumcenter over side also passes through this circle, under the above equivalence.

Let be the circumradius of the circle. In the diagram below

is the midpoint of and is the reflection of over , and so . Since and , we have:

In the diagram below, is the midpoint of , and so is a median in triangle .

So:

Similarly, is the midpoint of , and again is a median in triangle :

This follows from the two preceding examples, since we now have and .

#### (Main goal)

If equation (1) is satisfied, then from one of the equivalent statements here, the segment is perpendicuclar to the radius .

Since the quadrilateral is a parallelogram, it follows that it is a rectangle. Consequently, the points are concyclic. And the circle through these four points shares the same center with the nine-point center of the parent triangle.

Since is a diameter, the triangle is right-angled at . By one of the equivalent statements here, we conclude that equation (1) holds.

## Takeaway

In a *non-right* , let be the side-lengths, the orthocenter, the circumcenter, and the reflection of the circumcenter over side . Then the following statements are *equivalent*:

- is right-angled at
- is right-angled at
- quadrilateral is a rectangle
- points are concyclic with as diameter.

## Task

- (Half century) In a
*non-right*triangle , let be the side-lengths, the altitudes, the feet of the altitudes from the respective vertices, the circumradius, the circumcenter, the nine-point center, the orthocenter, the midpoint of side , and the reflection of over side . PROVE that the following*fifty*statements are*equivalent*:- 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
- 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
- segment is tangent to the nine-point circle at .

(Quite plenty, but can we reach ?)

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