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 . Then the following *seventy-five* 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 .

It will take quite some time to establish the above equivalence. In this post we select just five.

This follows from the Pythagorean identity .

By the extended law of sines, we have that . And so:

The converse also holds.

We use the identity and the extended sine law :

Since is non-right, take .

From example 3 we had . By the converse of example 1 . Thus .

## Takeaway

In a non-right triangle , let be the side-lengths and the circumradius. Then the following statements are *equivalent*:

More in the task below.

## 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 .

(Target reached! And surpassed!)

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