Last time we saw that the circle with diameter passes through vertex if and only if the equation below holds:

(1)

Let this circle also pass through vertex . Then the nine-point center has to be .

## Updated equivalence

We now add the following *four* equivalent statements to what we had at the end of our post on August 14:

- a circle with diameter passes through and
- the nine-point center coincides with vertex
- the reflection of over is
- the reflection of over is

Only triangle having satisfies these.

Since this circle passes through vertex , our previous post shows that equation (1) is satisfied:

Further, with as diameter, the two triangles and are right triangles with and .

By a result in this post, we conclude that the nine-point center is .

Easy.

Easy.

Easy.

## Usual example

The diagram below shows the circumcircle, the nine-point circle, and the circle with diameter .

Or this one:

## Takeaway

Consider with side-lengths , circumradius , circumcenter , orthocenter , and nine-point center . If equation (1) is satisfied, then the following statements are *equivalent*:

- a circle with diameter passes through and
- and are both equilateral
- is tangent to the circumcircle at
- the orthic triangle is equilateral
- the reflection of over is
- the reflection of over is
- is equilateral
- coincides with

These are some of the many equivalent descriptions of the isosceles triangle in which .

## Task

- (Early fifties) 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-two*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
- 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
- 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 .