If we now reflect the circumcenter over the sides of the triangle, the reflections do not lie on the circumcircle, unless a certain condition is met.

## Sixty degrees

In the presence of an interior angle of or , the reflection of the circumcenter lies on the circumcircle.

Let be the circumradius of the triangle, and let be its orthocenter. In the diagram below

suppose that actually lies on the circumcircle. Then must equal , right? We showed previously that in any triangle with orthocenter . But then always equals . And so:

As before, let’s denote by the reflection of the circumcenter over side . We’ll establish the result for the case.

The angle which the minor arc through and subtends at the center is equal to ; that is, *obtuse* , and so *reflex* . Note that is a *rhombus*, and so obtuse .

Since *reflex* , we conclude that lies on the circumcircle.

## Simple deduction

As a consequence of the preceding examples, we can characterize equilateral triangles via reflections of the circumcenter.

Easy.

## Sample diagrams

Note that this is a right triangle in which , , and . The circumcenter is the point , and its reflection over side is the point .

Here, , and . An extremely pleasant case.

## Takeaway

In , let be the circumcenter, and let be the reflection of over side . Then the following statements are *equivalent*:

- .

And a few more, if you view this post.

## Task

- (Late 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 , the reflection of over side , the reflection of over side , and the reflection of over side . PROVE that the following*fifty-seven*statements are*equivalent*:- 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
- 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 .