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# Reflecting the circumcenter II

If we reflect the orthocenter of a triangle over the sides of the triangle, the reflections lie on the circumcircle of the parent triangle.

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.

In any , let be the reflection of the circumcenter over side . If lies on the circumcircle, then either or .

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:

The proof can also be accomplished by using angle arguments. Particularly the fact that the angle an arc subtends at the center of a circle is twice the angle the arc subtends at any other part of the circumference.
In , suppose that (or that ). PROVE that the reflection of the circumcenter over side lies on the circumcircle.

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.

A triangle is equilateral if and only if the reflections of the circumcenter over the three sides ALL lie on the circumcircle of the triangle.

Easy.

## Sample diagrams

Consider with vertices at , , and .

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

Consider with vertices at , , and .

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:

1. .

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